# 3.1 Vector Algebra

## Detailed Theory: Vector Algebra

### **1. Basic Concepts of Vectors**

#### **1.1 What is a Vector?**

A vector is a mathematical object that has both **magnitude** (size) and **direction**.

**Examples from real life:**

* Force (push/pull in a specific direction)
* Velocity (speed with direction)
* Displacement (change in position)

#### **1.2 Scalar vs Vector**

**Scalar:** Only magnitude (no direction)

**Examples:** $$5$$ (a number), $$10$$ meters (distance), $$20^\circ C$$ (temperature)

**Vector:** Both magnitude and direction

**Examples:** $$5$$ meters North, $$10$$ m/s East, $$15$$ N downward

#### **1.3 Representation of Vectors**

**a) Graphical Representation**

A vector is shown as an arrow:

* **Length** represents magnitude
* **Arrowhead** shows direction

**b) Algebraic Representation**

In 2D: $$\vec{v} = a\hat{i} + b\hat{j}$$

In 3D: $$\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$$

where $$\hat{i}, \hat{j}, \hat{k}$$ are unit vectors along x, y, z axes.

**c) Component Form**

2D: $$\vec{v} = \langle a, b \rangle$$

3D: $$\vec{v} = \langle a, b, c \rangle$$

#### **1.4 Types of Vectors**

**a) Zero Vector (Null Vector)**

Magnitude = 0, direction undefined.

Notation: $$\vec{0}$$ or $$0$$

In component form: $$\langle 0, 0 \rangle$$ or $$\langle 0, 0, 0 \rangle$$

**b) Unit Vector**

Magnitude = 1.

Used to indicate direction only.

**Finding unit vector:** If $$\vec{v}$$ is a vector, its unit vector is:

$$
\hat{v} = \frac{\vec{v}}{|\vec{v}|}
$$

**c) Equal Vectors**

Two vectors are equal if they have same magnitude and same direction.

**d) Negative of a Vector**

Same magnitude but opposite direction.

If $$\vec{v} = \langle a, b, c \rangle$$, then $$-\vec{v} = \langle -a, -b, -c \rangle$$

**e) Position Vector**

Vector from origin to a point.

If point is $$P(x, y, z)$$, position vector is:

$$
\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}
$$

**f) Co-initial Vectors**

Vectors starting from same point.

**g) Collinear Vectors**

Vectors parallel to same line (or lying on same line).

***

### **2. Vector Operations**

#### **2.1 Addition of Vectors**

**a) Triangle Law**

If two vectors are represented as two sides of a triangle taken in order, their sum is represented by the third side.

**Graphically:** Place tail of second vector at head of first vector. The sum is vector from tail of first to head of second.

**b) Parallelogram Law**

If two vectors are represented as adjacent sides of a parallelogram, their sum is the diagonal through the common point.

**c) Component-wise Addition**

If $$\vec{a} = \langle a\_1, a\_2, a\_3 \rangle$$ and $$\vec{b} = \langle b\_1, b\_2, b\_3 \rangle$$, then:

$$
\vec{a} + \vec{b} = \langle a\_1 + b\_1, a\_2 + b\_2, a\_3 + b\_3 \rangle
$$

**d) Properties of Vector Addition**

1. **Commutative:** $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$
2. **Associative:** $$(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$$
3. **Additive Identity:** $$\vec{a} + \vec{0} = \vec{a}$$
4. **Additive Inverse:** $$\vec{a} + (-\vec{a}) = \vec{0}$$

#### **2.2 Subtraction of Vectors**

**a) Geometric Method**

$$\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$$

**b) Component Method**

If $$\vec{a} = \langle a\_1, a\_2, a\_3 \rangle$$ and $$\vec{b} = \langle b\_1, b\_2, b\_3 \rangle$$, then:

$$
\vec{a} - \vec{b} = \langle a\_1 - b\_1, a\_2 - b\_2, a\_3 - b\_3 \rangle
$$

#### **2.3 Scalar Multiplication**

**a) Definition**

If $$k$$ is a scalar and $$\vec{v}$$ is a vector, then $$k\vec{v}$$ is:

* Magnitude = $$|k| \times |\vec{v}|$$
* Direction: same as $$\vec{v}$$ if $$k > 0$$, opposite if $$k < 0$$

**b) Component Form**

If $$\vec{v} = \langle v\_1, v\_2, v\_3 \rangle$$, then:

$$
k\vec{v} = \langle kv\_1, kv\_2, kv\_3 \rangle
$$

**c) Properties**

1. $$k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$$
2. $$(k + l)\vec{a} = k\vec{a} + l\vec{a}$$
3. $$k(l\vec{a}) = (kl)\vec{a}$$
4. $$1 \cdot \vec{a} = \vec{a}$$

***

### **3. Magnitude and Direction**

#### **3.1 Magnitude (Length) of a Vector**

For $$\vec{v} = \langle a, b, c \rangle$$:

2D:

$$
|\vec{v}| = \sqrt{a^2 + b^2}
$$

3D:

$$
|\vec{v}| = \sqrt{a^2 + b^2 + c^2}
$$

#### **3.2 Direction Cosines**

For 3D vector $$\vec{v} = \langle a, b, c \rangle$$:

Direction cosines are:

$$
l = \cos\alpha = \frac{a}{|\vec{v}|}, \quad m = \cos\beta = \frac{b}{|\vec{v}|}, \quad n = \cos\gamma = \frac{c}{|\vec{v}|}
$$

where $$\alpha, \beta, \gamma$$ are angles with x, y, z axes.

**Property:**

$$
l^2 + m^2 + n^2 = 1
$$

#### **3.3 Direction Ratios**

Three numbers proportional to direction cosines.

If direction cosines are $$l, m, n$$, then direction ratios are $$a, b, c$$ where:

$$
l : m : n = a : b : c
$$

***

### **4. Dot Product (Scalar Product)**

#### **4.1 Definition**

The dot product of two vectors $$\vec{a}$$ and $$\vec{b}$$ is:

$$
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta
$$

where $$\theta$$ is angle between the vectors.

#### **4.2 Component Form**

If $$\vec{a} = \langle a\_1, a\_2, a\_3 \rangle$$ and $$\vec{b} = \langle b\_1, b\_2, b\_3 \rangle$$, then:

$$
\vec{a} \cdot \vec{b} = a\_1b\_1 + a\_2b\_2 + a\_3b\_3
$$

#### **4.3 Geometric Interpretation**

1. **Projection:** $$\vec{a} \cdot \vec{b} = |\vec{a}| \times (\text{projection of } \vec{b} \text{ on } \vec{a})$$
2. **Angle between vectors:**

$$
\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
$$

3. **Orthogonality:** $$\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$$ (vectors are perpendicular)

#### **4.4 Properties of Dot Product**

1. **Commutative:** $$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$$
2. **Distributive:** $$\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$$
3. **Scalar multiplication:** $$(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (k\vec{b})$$
4. **Self product:** $$\vec{a} \cdot \vec{a} = |\vec{a}|^2$$
5. $$\vec{a} \cdot \vec{a} = 0 \iff \vec{a} = \vec{0}$$

#### **4.5 Applications**

**a) Finding Angle Between Vectors**

**Example:** Find angle between $$\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$

**Solution:**

$$
\vec{a} \cdot \vec{b} = (1)(2) + (2)(-1) + (-1)(1) = 2 - 2 - 1 = -1
$$

$$
|\vec{a}| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}
$$

$$
|\vec{b}| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}
$$

$$
\cos\theta = \frac{-1}{\sqrt{6} \times \sqrt{6}} = \frac{-1}{6}
$$

$$
\theta = \cos^{-1}\left(-\frac{1}{6}\right)
$$

**b) Checking Perpendicularity**

**Example:** Are $$\vec{a} = \langle 2, 3, 1 \rangle$$ and $$\vec{b} = \langle 1, -1, 1 \rangle$$ perpendicular?

**Solution:**

$$
\vec{a} \cdot \vec{b} = (2)(1) + (3)(-1) + (1)(1) = 2 - 3 + 1 = 0
$$

Yes, they are perpendicular.

***

### **5. Cross Product (Vector Product)**

#### **5.1 Definition**

The cross product of two vectors $$\vec{a}$$ and $$\vec{b}$$ is:

$$
\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \ \hat{n}
$$

where:

* $$\theta$$ is angle between vectors
* $$\hat{n}$$ is unit vector perpendicular to both $$\vec{a}$$ and $$\vec{b}$$
* Direction determined by right-hand rule

#### **5.2 Right-Hand Rule**

Point fingers in direction of $$\vec{a}$$, curl toward $$\vec{b}$$, thumb points in direction of $$\vec{a} \times \vec{b}$$.

#### **5.3 Component Form**

If $$\vec{a} = \langle a\_1, a\_2, a\_3 \rangle$$ and $$\vec{b} = \langle b\_1, b\_2, b\_3 \rangle$$, then:

$$
\vec{a} \times \vec{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a\_1 & a\_2 & a\_3 \\
b\_1 & b\_2 & b\_3
\end{vmatrix}
$$

Expanding the determinant:

$$
\vec{a} \times \vec{b} = (a\_2b\_3 - a\_3b\_2)\hat{i} - (a\_1b\_3 - a\_3b\_1)\hat{j} + (a\_1b\_2 - a\_2b\_1)\hat{k}
$$

#### **5.4 Geometric Interpretation**

1. **Magnitude:** $$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta$$
2. **Area of parallelogram** with sides $$\vec{a}$$ and $$\vec{b}$$ = $$|\vec{a} \times \vec{b}|$$
3. **Area of triangle** with sides $$\vec{a}$$ and $$\vec{b}$$ = $$\frac{1}{2}|\vec{a} \times \vec{b}|$$
4. **Parallel vectors:** $$\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$$

#### **5.5 Properties of Cross Product**

1. **Anti-commutative:** $$\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$$
2. **Distributive:** $$\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$$
3. **Scalar multiplication:** $$(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b}) = \vec{a} \times (k\vec{b})$$
4. **Self product:** $$\vec{a} \times \vec{a} = \vec{0}$$
5. **Cross product with zero:** $$\vec{a} \times \vec{0} = \vec{0} \times \vec{a} = \vec{0}$$

#### **5.6 Applications**

**a) Finding Area**

**Example:** Find area of parallelogram with adjacent sides $$\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} - 2\hat{k}$$

**Solution:**

$$
\vec{a} \times \vec{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
2 & 1 & -1 \\
1 & 3 & -2
\end{vmatrix}
$$

$$
\= \hat{i}\[(1)(-2) - (-1)(3)] - \hat{j}\[(2)(-2) - (-1)(1)] + \hat{k}\[(2)(3) - (1)(1)]
$$

$$
\= \hat{i}\[-2 + 3] - \hat{j}\[-4 + 1] + \hat{k}\[6 - 1]
$$

$$
\= \hat{i}(1) - \hat{j}(-3) + \hat{k}(5) = \hat{i} + 3\hat{j} + 5\hat{k}
$$

Area = $$|\vec{a} \times \vec{b}| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}$$

**b) Finding Unit Perpendicular Vector**

**Example:** Find unit vector perpendicular to both $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$$

**Solution:**

First find $$\vec{a} \times \vec{b}$$:

$$
\vec{a} \times \vec{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 1 \\
2 & 1 & -1
\end{vmatrix}
$$

$$
\= \hat{i}\[(1)(-1) - (1)(1)] - \hat{j}\[(1)(-1) - (1)(2)] + \hat{k}\[(1)(1) - (1)(2)]
$$

$$
\= \hat{i}\[-1 - 1] - \hat{j}\[-1 - 2] + \hat{k}\[1 - 2]
$$

$$
\= \hat{i}(-2) - \hat{j}(-3) + \hat{k}(-1) = -2\hat{i} + 3\hat{j} - \hat{k}
$$

Magnitude = $$\sqrt{(-2)^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14}$$

Unit vector = $$\frac{-2\hat{i} + 3\hat{j} - \hat{k}}{\sqrt{14}}$$

***

### **6. Scalar Triple Product**

#### **6.1 Definition**

The scalar triple product of three vectors $$\vec{a}, \vec{b}, \vec{c}$$ is:

$$
\[\vec{a} \ \vec{b} \ \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})
$$

#### **6.2 Determinant Form**

If $$\vec{a} = \langle a\_1, a\_2, a\_3 \rangle$$, $$\vec{b} = \langle b\_1, b\_2, b\_3 \rangle$$, $$\vec{c} = \langle c\_1, c\_2, c\_3 \rangle$$, then:

$$
\[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix}
a\_1 & a\_2 & a\_3 \\
b\_1 & b\_2 & b\_3 \\
c\_1 & c\_2 & c\_3
\end{vmatrix}
$$

#### **6.3 Geometric Interpretation**

1. **Volume of parallelepiped** with edges $$\vec{a}, \vec{b}, \vec{c}$$ = $$|\[\vec{a} \ \vec{b} \ \vec{c}]|$$
2. **Volume of tetrahedron** with edges $$\vec{a}, \vec{b}, \vec{c}$$ = $$\frac{1}{6}|\[\vec{a} \ \vec{b} \ \vec{c}]|$$
3. **Coplanarity test:** $$\[\vec{a} \ \vec{b} \ \vec{c}] = 0 \iff \vec{a}, \vec{b}, \vec{c}$$ are coplanar

#### **6.4 Properties**

1. **Cyclic property:** $$\[\vec{a} \ \vec{b} \ \vec{c}] = \[\vec{b} \ \vec{c} \ \vec{a}] = \[\vec{c} \ \vec{a} \ \vec{b}]$$
2. **Anti-cyclic property:** $$\[\vec{a} \ \vec{b} \ \vec{c}] = -\[\vec{a} \ \vec{c} \ \vec{b}]$$
3. **Linear in each argument:** $$\[k\vec{a} \ \vec{b} \ \vec{c}] = k\[\vec{a} \ \vec{b} \ \vec{c}]$$
4. **If two vectors equal:** $$\[\vec{a} \ \vec{a} \ \vec{c}] = 0$$

#### **6.5 Applications**

**a) Checking Coplanarity**

**Example:** Check if vectors $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k}$$, $$\vec{c} = 7\hat{i} + 8\hat{j} + 9\hat{k}$$ are coplanar.

**Solution:**

Compute scalar triple product:

$$
\[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{vmatrix}
$$

Using Sarrus' rule:

* Left-to-right: $$(1)(5)(9) + (2)(6)(7) + (3)(4)(8) = 45 + 84 + 96 = 225$$
* Right-to-left: $$(3)(5)(7) + (1)(6)(8) + (2)(4)(9) = 105 + 48 + 72 = 225$$

Determinant = $$225 - 225 = 0$$

So vectors are coplanar.

**b) Finding Volume**

**Example:** Find volume of parallelepiped with edges $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$$, $$\vec{c} = \hat{i} - \hat{j} + \hat{k}$$

**Solution:**

$$
\[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix}
1 & 1 & 1 \\
2 & 3 & -1 \\
1 & -1 & 1
\end{vmatrix}
$$

Expand along first row:

$$
\= 1\begin{vmatrix}3 & -1 \ -1 & 1\end{vmatrix} - 1\begin{vmatrix}2 & -1 \ 1 & 1\end{vmatrix} + 1\begin{vmatrix}2 & 3 \ 1 & -1\end{vmatrix}
$$

$$
\= 1(3\cdot1 - (-1)(-1)) - 1(2\cdot1 - (-1)(1)) + 1(2(-1) - 3\cdot1)
$$

$$
\= 1(3 - 1) - 1(2 + 1) + 1(-2 - 3)
$$

$$
\= 2 - 3 - 5 = -6
$$

Volume = $$|\[\vec{a} \ \vec{b} \ \vec{c}]| = |-6| = 6$$

***

### **7. Vector Triple Product**

#### **7.1 Definition**

The vector triple product of three vectors $$\vec{a}, \vec{b}, \vec{c}$$ is:

$$
\vec{a} \times (\vec{b} \times \vec{c})
$$

#### **7.2 Important Identity (BAC-CAB Rule)**

$$
\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}
$$

**Mnemonic:** "BAC minus CAB"

#### **7.3 Properties**

1. **Not associative:** $$\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$$
2. **Jacobi identity:** $$\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = \vec{0}$$
3. **Result lies in plane** of $$\vec{b}$$ and $$\vec{c}$$

#### **7.4 Application**

**Example:** Simplify $$\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b})$$

**Solution:**

Using BAC-CAB rule:

First term: $$(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$$

Second term: $$(\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a}$$

Third term: $$(\vec{c} \cdot \vec{b})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$$

Add all terms:

From first: $$(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$$

From second: $$(\vec{a} \cdot \vec{b})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a}$$

From third: $$(\vec{b} \cdot \vec{c})\vec{a} - (\vec{a} \cdot \vec{c})\vec{b}$$

Sum = $$\vec{0}$$

***

### **8. Projection of Vectors**

#### **8.1 Scalar Projection**

Scalar projection of $$\vec{b}$$ onto $$\vec{a}$$:

$$
\text{comp}\_{\vec{a}}\vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}
$$

This is the signed length of projection.

#### **8.2 Vector Projection**

Vector projection of $$\vec{b}$$ onto $$\vec{a}$$:

$$
\text{proj}\_{\vec{a}}\vec{b} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right)\vec{a}
$$

This is the vector component of $$\vec{b}$$ in direction of $$\vec{a}$$.

#### **8.3 Orthogonal Component**

Component of $$\vec{b}$$ perpendicular to $$\vec{a}$$:

$$
\vec{b}*{\perp} = \vec{b} - \text{proj}*{\vec{a}}\vec{b}
$$

**Example:** Find projection of $$\vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k}$$ onto $$\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}$$

**Solution:**

$$
\vec{a} \cdot \vec{b} = (1)(4) + (2)(5) + (2)(6) = 4 + 10 + 12 = 26
$$

$$
|\vec{a}|^2 = 1^2 + 2^2 + 2^2 = 1 + 4 + 4 = 9
$$

$$
\text{proj}\_{\vec{a}}\vec{b} = \frac{26}{9}\vec{a} = \frac{26}{9}(\hat{i} + 2\hat{j} + 2\hat{k}) = \frac{26}{9}\hat{i} + \frac{52}{9}\hat{j} + \frac{52}{9}\hat{k}
$$

***

### **9. Lines in Space**

#### **9.1 Vector Equation of a Line**

Through point with position vector $$\vec{a}$$ and parallel to $$\vec{b}$$:

$$
\vec{r} = \vec{a} + \lambda\vec{b}, \quad \lambda \in \mathbb{R}
$$

where $$\vec{r}$$ is position vector of any point on line.

#### **9.2 Parametric Form**

If $$\vec{a} = \langle x\_1, y\_1, z\_1 \rangle$$ and $$\vec{b} = \langle a, b, c \rangle$$, then:

$$
x = x\_1 + \lambda a, \quad y = y\_1 + \lambda b, \quad z = z\_1 + \lambda c
$$

#### **9.3 Symmetric Form**

From parametric form (eliminating $$\lambda$$):

$$
\frac{x - x\_1}{a} = \frac{y - y\_1}{b} = \frac{z - z\_1}{c}
$$

#### **9.4 Line Through Two Points**

Through points with position vectors $$\vec{a}$$ and $$\vec{b}$$:

$$
\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})
$$

or

$$
\frac{x - x\_1}{x\_2 - x\_1} = \frac{y - y\_1}{y\_2 - y\_1} = \frac{z - z\_1}{z\_2 - z\_1}
$$

***

### **10. Planes in Space**

#### **10.1 Vector Equation of a Plane**

**a) Through point** $$\vec{a}$$ **and perpendicular to** $$\vec{n}$$**:**

$$
(\vec{r} - \vec{a}) \cdot \vec{n} = 0
$$

or

$$
\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}
$$

**b) Through three points** $$\vec{a}, \vec{b}, \vec{c}$$**:**

$$
(\vec{r} - \vec{a}) \cdot \[(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})] = 0
$$

or

$$
\[\vec{r} - \vec{a} \ \vec{b} - \vec{a} \ \vec{c} - \vec{a}] = 0
$$

#### **10.2 Cartesian Equation**

If $$\vec{n} = \langle A, B, C \rangle$$ and point is $$(x\_1, y\_1, z\_1)$$, then:

$$
A(x - x\_1) + B(y - y\_1) + C(z - z\_1) = 0
$$

or

$$
Ax + By + Cz + D = 0
$$

#### **10.3 Intercept Form**

If plane cuts x, y, z axes at $$a, b, c$$ respectively:

$$
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1
$$

#### **10.4 Distance from Point to Plane**

Point $$P(x\_0, y\_0, z\_0)$$ to plane $$Ax + By + Cz + D = 0$$:

$$
d = \frac{|Ax\_0 + By\_0 + Cz\_0 + D|}{\sqrt{A^2 + B^2 + C^2}}
$$

***

### **11. Important Formulas Summary**

#### **11.1 Basic Operations**

* Addition: $$\vec{a} + \vec{b} = \langle a\_1+b\_1, a\_2+b\_2, a\_3+b\_3 \rangle$$
* Subtraction: $$\vec{a} - \vec{b} = \langle a\_1-b\_1, a\_2-b\_2, a\_3-b\_3 \rangle$$
* Scalar multiplication: $$k\vec{a} = \langle ka\_1, ka\_2, ka\_3 \rangle$$
* Magnitude: $$|\vec{a}| = \sqrt{a\_1^2 + a\_2^2 + a\_3^2}$$

#### **11.2 Dot Product**

* $$\vec{a} \cdot \vec{b} = a\_1b\_1 + a\_2b\_2 + a\_3b\_3 = |\vec{a}||\vec{b}|\cos\theta$$
* Angle: $$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$
* Orthogonal: $$\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$$

#### **11.3 Cross Product**

*

\hat{i} & \hat{j} & \hat{k} \ a\_1 & a\_2 & a\_3 \ b\_1 & b\_2 & b\_3 \end{vmatrix}$$

* Magnitude: $$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$$
* Parallel: $$\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$$

#### **11.4 Scalar Triple Product**

*

a\_1 & a\_2 & a\_3 \ b\_1 & b\_2 & b\_3 \ c\_1 & c\_2 & c\_3 \end{vmatrix}$$

* Coplanar: $$\[\vec{a} \ \vec{b} \ \vec{c}] = 0 \iff \vec{a}, \vec{b}, \vec{c}$$ coplanar

#### **11.5 Projection**

* Scalar projection of $$\vec{b}$$ on $$\vec{a}$$: $$\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$$
* Vector projection of $$\vec{b}$$ on $$\vec{a}$$: $$\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right)\vec{a}$$

***

### **12. Solved Examples**

#### **Example 1:** Finding Unit Vector

Find unit vector in direction of $$\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$$

**Solution:**

$$
|\vec{a}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7
$$

Unit vector = $$\frac{\vec{a}}{|\vec{a}|} = \frac{2\hat{i} - 3\hat{j} + 6\hat{k}}{7} = \frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}$$

#### **Example 2:** Angle Between Vectors

Find angle between $$\vec{a} = \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$

**Solution:**

$$
\vec{a} \cdot \vec{b} = (1)(2) + (1)(-1) + (-1)(1) = 2 - 1 - 1 = 0
$$

Since dot product = 0, vectors are perpendicular. Angle = $$90^\circ$$

#### **Example 3:** Area of Triangle

Find area of triangle with vertices $$A(1,1,1)$$, $$B(2,3,5)$$, $$C(-1,0,2)$$

**Solution:**

Vectors:

$$
\vec{AB} = (2-1)\hat{i} + (3-1)\hat{j} + (5-1)\hat{k} = \hat{i} + 2\hat{j} + 4\hat{k}
$$

$$
\vec{AC} = (-1-1)\hat{i} + (0-1)\hat{j} + (2-1)\hat{k} = -2\hat{i} - \hat{j} + \hat{k}
$$

Area = $$\frac{1}{2}|\vec{AB} \times \vec{AC}|$$

$$
\vec{AB} \times \vec{AC} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & 2 & 4 \\
-2 & -1 & 1
\end{vmatrix}
$$

$$
\= \hat{i}\[(2)(1) - (4)(-1)] - \hat{j}\[(1)(1) - (4)(-2)] + \hat{k}\[(1)(-1) - (2)(-2)]
$$

$$
\= \hat{i}\[2 + 4] - \hat{j}\[1 + 8] + \hat{k}\[-1 + 4]
$$

$$
\= 6\hat{i} - 9\hat{j} + 3\hat{k}
$$

Magnitude = $$\sqrt{6^2 + (-9)^2 + 3^2} = \sqrt{36 + 81 + 9} = \sqrt{126}$$

Area = $$\frac{1}{2}\sqrt{126} = \frac{3\sqrt{14}}{2}$$

#### **Example 4:** Line Equation

Find vector equation of line through $$A(1,2,3)$$ and $$B(4,5,6)$$

**Solution:**

Position vectors: $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k}$$

Direction vector: $$\vec{b} - \vec{a} = 3\hat{i} + 3\hat{j} + 3\hat{k}$$

Equation: $$\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$$

$$
\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(3\hat{i} + 3\hat{j} + 3\hat{k})
$$

***

### **13. Exam Tips and Common Mistakes**

#### **13.1 Common Mistakes**

1. **Confusing dot and cross products:** Dot gives scalar, cross gives vector
2. **Direction of cross product:** Use right-hand rule correctly
3. **Unit vector calculation:** Forgetting to divide by magnitude
4. **Coplanarity test:** Remember $$\[\vec{a} \ \vec{b} \ \vec{c}] = 0$$ for coplanar vectors
5. **Parallel vectors:** $$\vec{a} \times \vec{b} = \vec{0}$$ for parallel vectors

#### **13.2 Problem-Solving Strategy**

1. **Identify what's given and what's asked**
2. **Choose appropriate formula** (dot, cross, triple product, etc.)
3. **Compute step by step**
4. **Check answer** for reasonableness
5. **Include units** if applicable

#### **13.3 Quick Checks**

1. **Dot product result** is always scalar
2. **Cross product result** is always vector perpendicular to both
3. **Scalar triple product** is scalar (volume)
4. **Vector triple product** is vector in plane of last two vectors
5. **Unit vector** has magnitude 1

This comprehensive theory covers all aspects of vector algebra with detailed explanations and examples, providing complete preparation for the entrance examination.