Vectors and Three-Dimensional Geometry

Definition

• Scalar: A quantity that has only magnitude (size/amount) but no direction.
• Vector: A quantity that has both magnitude and a specific direction.

Explanation

Imagine I tell you, "Walk 5 kilometers." You'll ask, "In which direction?" Because that information is missing, "5 km" is a scalar. If I say, "Walk 5 km North," I have given you a vector.

Key Points

• Scalars follow simple arithmetic (2kg + 3kg = 5kg).
• Vectors follow specific geometric rules (Triangle law, Parallelogram law).
• A vector is usually represented by an arrow $\vec{a}$.

Examples

• Scalar: Temperature (37°C), Mass (50kg), Time (10s).
• Vector: Velocity (60 km/h East), Force (10N downwards).

Real-Life Use

Navigation Systems: In a car's GPS, your speed is a scalar, but your "velocity" is a vector. The GPS needs to know both speed and heading to calculate arrival times.

Definition

• Magnitude: The length or size of the vector, denoted as $|\vec{a}|$.
• Position Vector: A vector that starts from the origin (0,0,0) and points to a specific location in space.
• Unit Vector: A vector with a magnitude of exactly 1, used purely to show direction.

Explanation

If you are at the center of a room (the origin) and you want to reach a corner, the path you take is your position vector. If you want to describe just the direction without caring how far it is, you use a unit vector.

Key Points

• Addition: Use the Head-to-Tail method.
• Scalar Multiplication: Multiplying by a number changes magnitude but keeps (or reverses) direction.
• Resolution: Breaking a vector into perpendicular components (x and y).

Examples

• Position Vector: Point (3, 4, 5) $\rightarrow \vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k}$.
• Scalar Multiplication: If $\vec{a} = 2\hat{i} + 3\hat{j}$, then $3\vec{a} = 6\hat{i} + 9\hat{j}$.

Real-Life Use

Structural Engineering: Engineers resolve force vectors into horizontal and vertical components to ensure bridges don't slide or collapse.

Definition

• Scalar (Dot) Product ($\vec{a} \cdot \vec{b}$): Result is a scalar. Formula: $|\vec{a}||\vec{b}|\cos\theta$.
• Vector (Cross) Product ($\vec{a} \times \vec{b}$): Result is a vector perpendicular to both. Formula: $|\vec{a}||\vec{b}|\sin\theta$.
• Scalar Triple Product: $\vec{a} \cdot (\vec{b} \times \vec{c})$, representing the volume of a parallelepiped.

Key Points

• $\vec{a} \cdot \vec{b} = 0$ if vectors are Perpendicular.
• $\vec{a} \times \vec{b} = 0$ if vectors are Parallel.
• Dot product is commutative; Cross product is not.

Comparison Table

Real-Life Use

Physics (Torque): Using a wrench. Torque is a Cross Product. Pulling parallel to the handle results in zero torque!

Definition

• Direction Cosines (DCs): Cosines of angles with x, y, z axes ($l, m, n$).
• Direction Ratios (DRs): Proportional numbers $(a, b, c)$ to the DCs.
• Equation of a Line: $\vec{r} = \vec{a} + \lambda\vec{b}$ (Vector) or $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ (Cartesian).
• Equation of a Plane: $\vec{r} \cdot \vec{n} = d$ or $Ax + By + Cz = D$.

Explanation

A Line is like a laser beam; you need a starting point and a direction. A Plane is like a floating sheet of paper; you need a point and a "normal" vector (a stick pointing straight out of it).

Real-Life Use

Video Game Design: Games like Minecraft use 3D geometry to calculate if a bullet (a line) hits a wall (a plane).

Essential Formulas

1. Magnitude: |a| = √(x² + y² + z²)
2. Unit Vector: â = a / |a|
3. Dot Product: a·b = a₁b₁ + a₂b₂ + a₃b₃
4. Shortest Distance (Skew Lines):
   d = |(a₂ - a₁) · (b₁ × b₂)| / |b₁ × b₂|
        

Numerical Problem 1: Finding the Angle

Question: Find the angle between $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{j} + \hat{k}$.

1. |a| = √(1² + 1² + 0²) = √2
2. |b| = √(0² + 1² + 1²) = √2
3. a·b = (1)(0) + (1)(1) + (0)(1) = 1
4. cosθ = 1 / (√2 * √2) = 1/2
5. θ = cos⁻¹(1/2) = 60°
        

Numerical Problem 2: Work Done

Question: Force $\vec{F} = 3\hat{i} + 2\hat{j} - 4\hat{k}$ moves a particle from $A(1,2,3)$ to $B(4,5,6)$. Find work done.

1. Displacement d = B - A = (3, 3, 3)
2. Work (W) = F · d
3. W = (3)(3) + (2)(3) + (-4)(3)
4. W = 9 + 6 - 12 = 3 units
        

Scenario: A drone flies from $P(0,0,0)$ to $D(10, 20, 30)$. Wind blows in direction $\vec{w} = -2\hat{i} + 1\hat{j} + 0\hat{k}$. A warehouse ceiling is a plane at $z = 40$.

• Flight Path: Position vector $\vec{d} = 10\hat{i} + 20\hat{j} + 30\hat{k}$.
• Wind Impact: Drone must apply an equal/opposite force $\vec{F}_{wind} = 2\hat{i} - 1\hat{j}$ to stay on track.
• Safety: Since max $z=30$ and ceiling is at $z=40$, the drone is safe by 10m.

1. If $\vec{a} \cdot \vec{b} = 0$, the angle between them is:
   Ans: c) 90°
2. The magnitude of the unit vector $\hat{i} + \hat{j} + \hat{k}$ is:
   Ans: c) √3
3. The cross product $\vec{a} \times \vec{a}$ is always:
   Ans: c) $\vec{0}$
4. Direction cosines $l, m, n$ satisfy:
   Ans: c) $l^2+m^2+n^2=1$
5. The dot product of $\vec{a} = 2\hat{i}$ and $\vec{b} = 3\hat{j}$ is:
   Ans: c) 0

1. Q: Prove $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$ are perpendicular.
   Ans: Show $\vec{a} \cdot \vec{b} = (1)(2) + (-2)(1) + (3)(-1) \neq 0$ (Wait, check math: $2-2-3 = -3$. If dot product isn't 0, they aren't perpendicular. Always check calculation!)
2. Q: Find the area of triangle with vertices A(1,1,1), B(1,2,3), C(2,3,1).
   Ans: Area = $\frac{1}{2} |\vec{AB} \times \vec{AC}|$.
3. Q: Find the distance of point (2,3,4) from plane $x-y+z=5$.
   Ans: Use $d = \frac{|2-3+4-5|}{\sqrt{1^2+(-1)^2+1^2}} = \frac{2}{\sqrt{3}}$.

• Dot vs. Cross: Dot is Dull (Scalar). Cross is Cool (Vector).
• DCs vs. DRs: Cosine is Complete (Unit values). Ratios are Relative (Proportions).
• STP: Scalar Triple Product = Space-Triple-Product (Volume).

Recommended Tools: Use GeoGebra 3D for visualization and WolframAlpha for step-by-step math verification.