Chapter 12 – Vector Algebra (JEE)
1. Introduction to Vectors
A vector is a quantity that has both magnitude and direction.
Examples: displacement, velocity, force.
Quantities having only magnitude are called scalars.
2. Representation of Vectors
Vectors are represented by directed line segments.
Length → magnitude, arrow → direction.
Vector $\vec{AB}$ represents displacement from point $A$ to point $B$.
3. Types of Vectors
| Vector Type | Description |
|---|---|
| Zero Vector | Magnitude zero |
| Unit Vector | Magnitude one |
| Position Vector | Vector from origin to a point |
| Equal Vectors | Same magnitude and direction |
| Negative Vectors | Same magnitude, opposite direction |
4. Magnitude of a Vector
If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, then
$$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$
5. Unit Vector
Unit vector in direction of $\vec{a}$:
$$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$$
6. Position Vector
Position vector of point $P(x,y,z)$ is:
$\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$
7. Vector Addition
Two vectors are added using triangle law or parallelogram law.
$\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$
8. Scalar Multiplication
Multiplying a vector by a scalar changes magnitude but not direction.
$k\vec{a}$ has magnitude $|k||\vec{a}|$
9. Direction Cosines and Ratios
If vector makes angles $\alpha,\beta,\gamma$ with axes:
$$\cos\alpha = \frac{a_1}{|\vec{a}|}$$
Direction ratios are proportional to direction cosines.
10. Scalar (Dot) Product
$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$
Dot product is a scalar quantity.
11. Properties of Dot Product
- $\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$
- $\vec{a}\cdot\vec{a} = |\vec{a}|^2$
- If $\vec{a}\cdot\vec{b} = 0$, vectors are perpendicular
12. Projection of a Vector
Projection of $\vec{a}$ on $\vec{b}$:
$$\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$$
13. Vector (Cross) Product
$\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$
Direction given by right-hand thumb rule.
14. Properties of Cross Product
- $\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})$
- $\vec{a}\times\vec{a} = 0$
- Parallel vectors → cross product zero
15. Area Using Cross Product
Area of parallelogram = $|\vec{a}\times\vec{b}|$
Area of triangle = $\frac12 |\vec{a}\times\vec{b}|$
16. Scalar Triple Product
$[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$
17. Geometrical Meaning of Scalar Triple Product
Magnitude gives volume of parallelepiped formed by three vectors.
If scalar triple product is zero → vectors are coplanar.
18. Vector Triple Product
$\vec{a}\times(\vec{b}\times\vec{c})
= \vec{b}(\vec{a}\cdot\vec{c}) - \vec{c}(\vec{a}\cdot\vec{b})$
19. Coplanarity Condition
$[\vec{a}\ \vec{b}\ \vec{c}] = 0$
20. Common JEE Traps
- Confusing dot and cross product
- Forgetting direction in cross product
- Ignoring magnitude sign
- Wrong application of projection formula
21. Final Revision Checklist
You have mastered Vector Algebra if you can:
- Find magnitude and unit vector
- Apply dot and cross product correctly
- Use scalar triple product for volume and coplanarity
- Solve geometry-based vector problems