Vectors and Three-Dimensional Geometry

Meaning

In daily life, we measure many things. Some things only need a size, while others need a direction too. A scalar quantity is a physical quantity that has magnitude only, with no specific direction. A vector quantity is a physical quantity that has both magnitude and direction, and obeys the triangle law of addition.

Definitions

• According to classic mathematical physics texts, a scalar is defined as a quantity that is completely specified by a single real number along with a suitable unit of measurement.
• Erwin Kreyszig defines a vector in his engineering mathematics text as a directed line segment representing a quantity that possesses both magnitude and direction.

Explanation

Think of walking to the canteen. If I tell you I walked five meters, you do not know where I ended up. That is a scalar. But if I tell you I walked five meters towards the North, you now know my exact displacement. That is a vector.

Key Points

• Scalars have only magnitude (e.g., mass, temperature, time, speed).
• Vectors have magnitude and direction (e.g., displacement, velocity, acceleration, force).
• Geometrically, a vector is represented by a directed line segment with an initial point and a terminal point.

Examples

• Mass of a bag of rice is 10 kilograms (Scalar).
• Wind is blowing at 40 kilometers per hour towards the East (Vector).

Meaning

The magnitude of a vector is its length or absolute value, while its direction is the spatial orientation of the line along which it acts.

Definitions

• Mathematical textbooks define the magnitude of a vector as the Euclidean distance between its initial and terminal points.
• Direction is defined by the angles the vector makes with the positive coordinate axes.

Explanation

Imagine an arrow drawn on paper. The length of that arrow is its magnitude. Where the arrowhead is pointing tells you its direction. If a vector is given in component form as \(a\hat{i} + b\hat{j} + c\hat{k}\), its magnitude is simply the square root of the sum of the squares of its components.

Key Points

• Magnitude is always a non-negative real number.
• Represented mathematically as the absolute value symbol around the vector name.
• A unit vector has a magnitude of exactly one and is used solely to designate direction.

Examples

• The vector \(3\hat{i} + 4\hat{j}\) has a magnitude of the square root of \(3^2 + 4^2\), which equals 5 units.
• A force of 10 Newtons acting vertically upwards.

Meaning

Vectors are categorized into various types based on the relationship between their magnitudes, directions, and initial positions.

Definitions

• A zero vector is defined as a vector having zero magnitude and an indeterminate direction.
• Like vectors are defined as vectors having the same direction regardless of their magnitudes.

Explanation

Vectors come in different shapes and sizes based on how they behave. Some start at a fixed point, some can move freely, and some point in exact opposite directions.

Key Points

• Zero Vector: Initial and terminal points coincide (\(0\)).
• Unit Vector: Magnitude is 1 (\(\hat{a} = \frac{\vec{a}}{|\vec{a}|}\)).
• Collinear Vectors: Parallel to the same line, regardless of magnitude.
• Coplanar Vectors: Vectors lying in the same plane or parallel to the same plane.
• Equal Vectors: Same magnitude and same direction.

Examples

• The position vectors of points moving along a straight track.
• The unit vector pointing purely along the x-axis, written as \(\hat{i}\).

Meaning

Vector addition is the process of combining two or more vectors to yield a single resultant vector, while scalar multiplication scales a vector's magnitude without altering its line of action.

Definitions

• The triangle law of vector addition states that if two vectors are represented by two consecutive sides of a triangle in magnitude and direction, their sum is represented by the third side taken in reverse order.
• Scalar multiplication of a vector by a real number multiplies its magnitude by the absolute value of that number.

Explanation

If you walk 3 meters east and then 4 meters north, your net displacement is the hypotenuse of that triangle. That is vector addition. If you take a vector and multiply it by 2, you make it twice as long in the same direction. If you multiply by negative 2, it becomes twice as long and flips backward.

Key Points

• Triangle Law and Parallelogram Law govern vector addition.
• Vector addition is commutative (\(\vec{a} + \vec{b} = \vec{b} + \vec{a}\)) and associative.
• Multiplying by a scalar changes length and can reverse direction if negative.

Examples

• Adding wind velocity to an airplane's engine velocity to find its actual ground track.
• Doubling a force vector from 5 Newtons to 10 Newtons by multiplying by the scalar 2.

Meaning

A position vector specifies the location of a point relative to an origin in space, and the resolution of a vector breaks a single vector down into its mutually perpendicular components along the coordinate axes.

Definitions

• A position vector is defined as a vector drawn from a fixed origin point to a point in space.
• Resolution of a vector is the process of splitting a single vector into two or more component vectors along specific reference axes.

Explanation

Imagine standing in a room. If you treat the corner of the floor as the origin \((0,0,0)\), your exact position relative to that corner is your position vector. Any vector in three-dimensional space can be broken down into three parts: how much it goes along the x-axis, how much along the y-axis, and how much along the z-axis.

Key Points

• Position vector of point \(P(x,y,z)\) is \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\).
• Resolution lets us deal with complex space problems using simple algebra.
• Components simplify calculations drastically.

Examples

• A drone hovering at coordinates \((2, 3, 5)\) has a position vector of \(2\hat{i} + 3\hat{j} + 5\hat{k}\).
• Breaking a ramp sliding force into horizontal and vertical components.

Meaning

The scalar dot product multiplies two vectors to produce a scalar number representing how aligned they are, while the vector cross product multiplies two vectors to produce a third vector perpendicular to both.

Definitions

• The scalar product of two vectors \(\vec{a}\) and \(\vec{b}\) is defined as \(|\vec{a}| |\vec{b}| \cos\theta\), where \(\theta\) is the angle between them.
• The vector product of two vectors \(\vec{a}\) and \(\vec{b}\) is defined as \(|\vec{a}| |\vec{b}| \sin\theta \, \hat{n}\), where \(\hat{n}\) is a unit vector perpendicular to both.

Explanation

Think of the dot product as checking teamwork: are these two vectors pulling in the same direction? If they are perpendicular, their dot product is zero. The cross product, on the other hand, measures rotation or perpendicular span. If you take two vectors forming two sides of a parallelogram, their cross product gives a vector whose magnitude is the total area of that parallelogram.

Key Points

• Dot product: \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\). Yields a scalar.
• Cross product: Evaluated using a determinant with \(\hat{i}, \hat{j}, \hat{k}\). Yields a vector.
• If dot product is zero, vectors are perpendicular.
• If cross product is zero, vectors are parallel.

Examples

• Calculating work done when force and displacement are at an angle.
• Finding the torque vector generated by a wrench pushing on a bolt.

Meaning

The scalar triple product combines three vectors via a dot and cross product to yield a scalar volume, while work and moment apply these vector products to physical forces.

Definitions

• The scalar triple product of three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is defined as \(\vec{a} \cdot (\vec{b} \times \vec{c})\), representing the volume of a parallelepiped.
• Work done is the scalar product of force and displacement vectors. Moment of a force is the cross product of the position vector and the force vector.

Explanation

Imagine stacking sheets of paper into a slanted box shape called a parallelepiped. The scalar triple product tells you the exact volume of that box using three edge vectors. In physics, when you push a heavy box across the floor at an angle, the work done is just the dot product of your pushing force and the displacement. If you turn a rusty nut with a spanner, the turning power, or moment, is the cross product of your arm distance and the applied force.

Key Points

• Scalar Triple Product = Determinant formed by the components of the three vectors.
• If scalar triple product is zero, the three vectors are coplanar.
• Work Done \(W = \vec{F} \cdot \vec{d}\).
• Moment \(\vec{\tau} = \vec{r} \times \vec{F}\).

Examples

• Calculating the volume of a crystalline unit cell in chemistry using three edge vectors.
• Finding the torque applied when tightening a bicycle pedal.

Meaning

Direction cosines are the cosines of the angles made by a line with the positive axes, while direction ratios are any numbers proportional to these direction cosines.

Definitions

• Direction cosines of a line are the cosines of the angles \(\alpha, \beta, \gamma\) that the line makes with the positive \(x\), \(y\), and \(z\) axes respectively, denoted by \(l, m, n\).
• Direction ratios are numbers \(a, b, c\) such that \(\frac{a}{l} = \frac{b}{m} = \frac{c}{n}\).

Explanation

Imagine holding a pencil floating in the middle of your room. How is it tilted? If you measure the angle it makes with the x-axis, y-axis, and z-axis, and take the cosine of those angles, you get its direction cosines. Often, working with fractions is annoying, so we use simpler proportional whole numbers called direction ratios.

Key Points

• \(l^2 + m^2 + n^2 = 1\).
• Direction ratios \(a, b, c\) are converted to direction cosines using: \(l = \frac{a}{\pm\sqrt{a^2+b^2+c^2}}\), and similarly for \(m\) and \(n\).
• They define the exact orientation of any line in space.

Examples

• The direction cosines of the x-axis are \(1, 0, 0\) because it makes angles of \(0^\circ, 90^\circ, 90^\circ\) with the axes.
• A line with direction ratios \(2, -1, 2\).

Meaning

The equation of a line in space defines every point residing on that straight line using a point and a direction, and the angle between lines is the angle formed by their directional vectors.

Definitions

• The vector equation of a line passing through a point with position vector \(\vec{a}\) and parallel to vector \(\vec{b}\) is given by \(\vec{r} = \vec{a} + \lambda\vec{b}\).
• The Cartesian equation of a line is given by \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\).

Explanation

To draw a straight line in your room, you need two things: a starting point and a direction in which you are walking. That is all the vector equation of a line is. \(\vec{a}\) is your starting point, and \(\vec{b}\) is the direction you walk. To find the angle between two lines in space, you simply look at the angle between their direction vectors using the dot product formula.

Key Points

• Vector form: \(\vec{r} = \vec{a} + \lambda\vec{b}\).
• Cartesian form: \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\).
• Angle \(\theta\) between two lines with direction ratios \((a_1,b_1,c_1)\) and \((a_2,b_2,c_2)\) is found via: \(\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\).

Examples

• Finding the trajectory line of a bullet fired from point \((1, 2, 3)\) with direction ratios \(2, 2, 1\).
• Finding the angle between two intersecting alleyways represented as intersecting lines.

Meaning

The shortest distance is the length of the unique line segment that is mutually perpendicular to two skew lines in space.

Definitions

• The shortest distance between two skew lines is defined as the length of the perpendicular dropped between them.

Explanation

Imagine two airplanes flying at different altitudes on paths that do not cross and are not parallel. They are flying on skew lines. If you want to drop a measuring tape straight down from the top plane to touch the bottom plane's flight path at a 90-degree angle, the length of that tape is the shortest distance between those two lines.

Key Points

• For skew lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\), the shortest distance formula is: \(d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}\).
• For parallel lines, the formula simplifies because their direction vectors are identical.

Examples

• Calculating the clearance distance between two high-voltage power lines crossing each other at different heights on utility poles.
• Ensuring two roller coaster tracks never come closer than a safe minimum threshold.

Meaning

A plane is a flat, two-dimensional surface extending infinitely in three-dimensional space, defined by a normal vector and a point, and we can calculate angles between planes as well as distances from points to planes.

Definitions

• The vector equation of a plane normal to vector \(\vec{n}\) and passing through a point with position vector \(\vec{a}\) is \((\vec{r} - \vec{a}) \cdot \vec{n} = 0\).
• The Cartesian equation of a plane is given by \(Ax + By + Cz + D = 0\).

Explanation

Think of a flat tabletop floating in your room. That tabletop is a plane. To describe it mathematically, you need a point on the table and a flashlight beam standing straight up off the table at a 90-degree angle. That upright beam is your normal vector. Once you have that normal vector \((A, B, C)\), writing the equation of the plane is as simple as \(Ax + By + Cz + D = 0\).

Key Points

• Normal vector coefficients \(A, B, C\) in \(Ax + By + Cz + D = 0\) give the direction perpendicular to the plane.
• Angle between two planes is the angle between their normal vectors.
• Distance \(d\) from a point \((x_1, y_1, z_1)\) to a plane \(Ax + By + Cz + D = 0\) is: \(d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}\).

Examples

• Finding the equation of a flat wall in a building.
• Calculating the distance of a ceiling lamp from the floor plane.

Meaning

The angle between a line and a plane is the acute angle formed by the line and its orthogonal projection onto that plane.

Definitions

• The angle \(\theta\) between a line with direction vector \(\vec{b}\) and a plane with normal vector \(\vec{n}\) is given by \(\sin\theta = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}| |\vec{n}|}\).

Explanation

Imagine a spear being thrown at a slanted roof. The spear is your line, and the roof is your plane. The angle the spear makes with the roof is what we are looking for. Notice that the normal vector stands straight up from the roof at \(90^\circ\). Because of this geometry, we use the sine function instead of cosine when calculating this angle using the line's direction vector and the plane's normal vector.

Key Points

• Formula uses \(\sin\theta\), unlike lines and planes which use \(\cos\theta\).
• If the line is parallel to the plane, the angle is zero.
• If the line is perpendicular to the plane, the angle is \(90^\circ\) and \(\vec{b}\) is parallel to \(\vec{n}\).

Examples

• Calculating the angle at which solar rays strike a tilted solar panel mounted on a slanted roof.
• Finding the landing trajectory angle of a spacecraft touching down on a sloping planetary surface.

Quick Reference Table

Problem

Find the shortest distance between the two skew lines given by the vector equations:
\(\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})\)
and
\(\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(3\hat{i} + 4\hat{j} + 5\hat{k})\)

Step-by-Step Solution

Step 1: Identify the position vectors and direction vectors.
From line 1: \(\vec{a}_1 = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b}_1 = 2\hat{i} + 3\hat{j} + 4\hat{k}\)
From line 2: \(\vec{a}_2 = 2\hat{i} + 4\hat{j} + 5\hat{k}\), \(\vec{b}_2 = 3\hat{i} + 4\hat{j} + 5\hat{k}\)

Step 2: Calculate vector difference (\(\vec{a}_2 - \vec{a}_1\)).
\((\vec{a}_2 - \vec{a}_1) = (2-1)\hat{i} + (4-2)\hat{j} + (5-3)\hat{k} = 1\hat{i} + 2\hat{j} + 2\hat{k}\)

Step 3: Calculate cross product (\(\vec{b}_1 \times \vec{b}_2\)).
\(\vec{b}_1 \times \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{vmatrix}\)
\(= \hat{i}(15 - 16) - \hat{j}(10 - 12) + \hat{k}(8 - 9) = -\hat{i} + 2\hat{j} - \hat{k}\)

Step 4: Find the magnitude of (\(\vec{b}_1 \times \vec{b}_2\)).
\(|\vec{b}_1 \times \vec{b}_2| = \sqrt{(-1)^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}\)

Step 5: Calculate dot product of difference and cross product.
\((\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = (1)(-1) + (2)(2) + (2)(-1) = -1 + 4 - 2 = 1\)

Step 6: Apply shortest distance formula.
\(d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|} = \frac{1}{\sqrt{6}}\) units.
        

Applications

• CAD Software (SolidWorks, AutoCAD, Blender): Use vector resolution and plane equations extensively to model three-dimensional mechanical parts, render lighting, and simulate physical impacts.
• Robotics Operating System (ROS): Autonomous robots use position vectors, coordinate transformation matrices, and quaternion rotations to navigate complex rooms.
• Geographic Information Systems (GIS): Used by mapping services like Google Maps to calculate optimal GPS navigation vectors across curved topological earth planes.
• Finite Element Analysis (FEA): Calculates stress tensors, forces, and moments across thousands of interconnected vector meshes to test structural integrity.

Scenario & Analysis

A robotics startup is designing an automated robotic crane arm for an automated warehouse assembly line. The robotic arm lifts a heavy engine block from a conveyor belt, rotates it, and places it onto a mounting chassis plate. Engineers use a 3D coordinate system anchored at \((0, 0, 0)\).

Displacement vector: \(\vec{d} = 2\hat{i} + 4\hat{j} + 4\hat{k}\) meters. Work done by lifting force \(\vec{F} = 50\hat{i} + 100\hat{j} + 200\hat{k}\) is \(W = \vec{F} \cdot \vec{d} = 1300\) Joules. Rotational torque moment vector is \(\vec{\tau} = 900\hat{i} - 550\hat{j} + 50\hat{k}\) N-m.

1. What is the magnitude of the vector \(2\hat{i} - 3\hat{j} + 6\hat{k}\)?
   A) 5
   B) 7
   C) 49
   D) 11
   Answer: B
   Explanation: Magnitude is calculated as \(\sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7\).
2. If two vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other, then their scalar dot product is:
   A) 1
   B) -1
   C) 0
   D) Undefined
   Answer: C
   Explanation: The dot product is \(|\vec{a}||\vec{b}|\cos\theta\). Since \(\cos(90^\circ) = 0\), the dot product is 0.
3. The cross product of two parallel vectors is always:
   A) A unit vector
   B) A zero vector
   C) Equal to their dot product
   D) 1
   Answer: B
   Explanation: Since \(\sin(0^\circ) = 0\), the magnitude of the cross product is zero.
4. What is the sum of the squares of the direction cosines (\(l^2 + m^2 + n^2\)) of any line in space?
   A) 0
   B) 0.5
   C) 1
   D) Infinite
   Answer: C
   Explanation: The sum of squares of direction cosines is always identically equal to 1.
5. The angle \(\theta\) between two lines with direction ratios \((1, 2, 2)\) and \((3, 0, 4)\) is found using:
   A) Cross product
   B) Dot product formula with cosine
   C) Scalar triple product
   D) Point-plane distance formula
   Answer: B
   Explanation: The angle between two lines uses the dot product of direction ratios combined with the cosine formula.
6. What does the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) geometrically represent?
   A) Area of a triangle
   B) Length of a line
   C) Volume of a parallelepiped
   D) Work done by a force
   Answer: C
   Explanation: It computes the determinant whose absolute value gives the geometric volume of a parallelepiped.
7. The normal vector to the plane given by the equation \(3x - 4y + 5z = 12\) has components:
   A) \((3, -4, 5)\)
   B) \((12, 12, 12)\)
   C) \((3, 4, 5)\)
   D) \((-3, 4, -5)\)
   Answer: A
   Explanation: The coefficients of \(x, y\), and \(z\) directly represent components of the normal vector.
8. Work done by a constant force vector \(\vec{F}\) acting through a displacement vector \(\vec{d}\) is calculated as:
   A) \(\vec{F} \times \vec{d}\)
   B) \(\vec{F} \cdot \vec{d}\)
   C) \(|\vec{F}| / |\vec{d}|\)
   D) \((\vec{F} \cdot \vec{d}) \hat{i}\)
   Answer: B
   Explanation: Work done is physically defined as the scalar dot product of force and displacement vectors.
9. If three vectors lie in the same plane or are parallel to the same plane, they are called:
   A) Collinear
   B) Coplanar
   C) Orthogonal
   D) Unit vectors
   Answer: B
   Explanation: Vectors sharing a common flat plane are termed coplanar vectors.
10. What formula is used to calculate the angle between a line and a plane?
    A) Cosine formula
    B) Sine formula
    C) Tangent formula
    D) Pythagorean theorem
    Answer: B
    Explanation: Using the normal vector of a plane yields a sine relationship rather than cosine for a line-plane angle.

1. Define a position vector. If point \(P\) has coordinates \((3, 4, 5)\), write its position vector and find its magnitude.
   Answer: A position vector is drawn from a fixed origin to a point in space. For \(P(3, 4, 5)\), \(\vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k}\). Magnitude \(|\vec{r}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} = 5\sqrt{2}\) units.
2. Prove that vectors \(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}\), and \(\vec{c} = 3\hat{i} - 4\hat{j} - 4\hat{k}\) form the sides of a right-angled triangle.
   Answer: \(\vec{a} + \vec{b} = \vec{c}\), forming a triangle. Dot product \(\vec{a} \cdot \vec{b} = (2)(1) + (-1)(-3) + (1)(-5) = 2 + 3 - 5 = 0\), proving they are perpendicular and form a right-angled triangle.
3. Find the unit vector in the direction of \(\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}\).
   Answer: Magnitude \(|\vec{a}| = \sqrt{1 + 4 + 4} = 3\). Unit vector \(\hat{a} = \frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\).
4. Find the projection of vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
   Answer: Projection = \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{2(1) + 3(2) + 2(1)}{\sqrt{1^2 + 2^2 + 1^2}} = \frac{10}{\sqrt{6}} = \frac{5\sqrt{6}}{3}\).
5. Find the angle between vectors \(\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}\) and \(\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}\).
   Answer: \(\cos\theta = \frac{3 + 4 + 3}{\sqrt{14}\sqrt{14}} = \frac{10}{14} = \frac{5}{7}\). \(\theta = \cos^{-1}\left(\frac{5}{7}\right)\).
6. Find the Cartesian equation of the line through \((1, 2, 3)\) parallel to \(3\hat{i} + 4\hat{j} - 5\hat{k}\).
   Answer: \(\frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - 3}{-5}\).
7. Find the distance of the point \((1, -2, 3)\) from the plane \(2x - y + 2z + 3 = 0\).
   Answer: \(d = \frac{|2(1) - (-2) + 2(3) + 3|}{\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{|13|}{3} = \frac{13}{3}\) units.
8. Find the vector equation of a plane through \(2\hat{i} - \hat{j} + \hat{k}\) normal to \(4\hat{i} + 2\hat{j} - 3\hat{k}\).
   Answer: \((\vec{r} - (2\hat{i} - \hat{j} + \hat{k})) \cdot (4\hat{i} + 2\hat{j} - 3\hat{k}) = 0 \implies \vec{r} \cdot (4\hat{i} + 2\hat{j} - 3\hat{k}) = 3\).
9. Calculate work done by force \(\vec{F} = 3\hat{i} + 2\hat{j} - 5\hat{k}\) displacing a particle from \(A(1, 2, -3)\) to \(B(2, -1, 4)\).
   Answer: Displacement \(\vec{d} = 1\hat{i} - 3\hat{j} + 7\hat{k}\). Work done \(W = (3)(1) + (2)(-3) + (-5)(7) = -38\) Joules.
10. Find the angle between planes \(2x + y - 2z = 5\) and \(3x - 6y - 2z = 7\).
    Answer: \(\cos\theta = \frac{6 - 6 + 4}{(3)(7)} = \frac{4}{21} \implies \theta = \cos^{-1}\left(\frac{4}{21}\right)\).

• Dot vs. Cross: Dot is Cos (\(ab\cos\theta\)), Cross is Sin (\(ab\sin\theta\hat{n}\)).
• Line vs. Plane Angles: Lines use Cosine formulas, Planes use Sine formulas.
• Direction Cosines: \(l^2 + m^2 + n^2 = 1\).
• Cross Product Determinant: Remember the "+, -, +" sign alternation for \(\hat{i}, \hat{j}, \hat{k}\).

Mnemonic: "CoCos and SinCross" keeps your dot and cross product trigonometry straight!