Mathematics: Class 12 (Book 2) — Chapter 4: Vector Algebra

• Scalar Quantities: Quantities involving only magnitude (real numbers). Examples: length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance.
• Vector Quantities: Quantities involving both magnitude and direction. Examples: displacement, velocity, acceleration, force, weight, momentum, electric field intensity.
• Directed Line: A straight line $l$ in a plane or 3D space with a prescribed direction indicated by arrowheads.
• Directed Line Segment: A portion of a directed line restricted to a segment $AB$. It has a definite magnitude (length) and direction.
• Vector: A quantity that has magnitude as well as direction. Denoted as $\vec{AB}$ or $\vec{a}$.
• Initial Point: The point $A$ where the vector starts.
• Terminal Point: The point $B$ where the vector ends.
• Magnitude: The length of the vector, denoted as $|\vec{AB}|$, $|\vec{a}|$, or $a$. Since length is non-negative, $|\vec{a}| \ge 0$.

Consider a point $P(x, y, z)$ in a three-dimensional rectangular coordinate system with origin $O(0, 0, 0)$. The vector $\vec{OP}$ is called the position vector of point $P$ with respect to $O$, denoted as $\vec{r}$.

Using the distance formula, the magnitude of $\vec{OP}$ (or $\vec{r}$) is given by:

Let $\alpha, \beta, \gamma$ be the angles (called direction angles ) made by the position vector $\vec{r}$ with the positive directions of the $x, y,$ and $z$-axes respectively.

• Direction Cosines: The cosine values of these angles, i.e., $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.
• Coordinates of P: Expressed as $(lr, mr, nr)$ where $r = |\vec{r}|$.
• Direction Ratios: The numbers $a = lr$, $b = mr$, $c = nr$, which are proportional to the direction cosines, denoted as $a, b, c$.
• Fundamental Identity: $l^2 + m^2 + n^2 = 1$, but in general, $a^2 + b^2 + c^2 \neq 1$.

If two vectors are represented by two sides of a triangle taken in order, then their sum (resultant) is represented by the third side of the triangle taken in the opposite order.

$$\vec{AB} + \vec{BC} = \vec{AC}$$

When the sides of a triangle are taken in order, the resultant is zero: $\vec{AB} + \vec{BC} + \vec{CA} = \vec{AA} = \vec{0}$.

If two vectors $\vec{a}$ and $\vec{b}$ are represented by two adjacent sides of a parallelogram in magnitude and direction, then their sum $\vec{a} + \vec{b}$ is represented by the diagonal of the parallelogram passing through their common point.

$$\vec{OA} + \vec{OB} = \vec{OC} \quad \text{(since } \vec{AC} = \vec{OB}\text{)}$$

• Commutative Property: For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.
• Associative Property: For any three vectors $\vec{a}, \vec{b},$ and $\vec{c}$, $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$.
• Additive Identity: The zero vector $\vec{0}$ acts as the identity: $\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}$.

Let $\hat{i}, \hat{j}, \hat{k}$ be the unit vectors along the positive $x, y, z$ axes respectively.

Any vector $\vec{r}$ can be expressed in its component form as:

• Scalar Components: $x, y, z$ (also called direction ratios).
• Vector Components: $x\hat{i}, y\hat{j}, z\hat{k}$.
• Magnitude: $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$.

Let $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$:

• Sum: $\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$
• Difference: $\vec{a} - \vec{b} = (a_1-b_1)\hat{i} + (a_2-b_2)\hat{j} + (a_3-b_3)\hat{k}$
• Equality: $\vec{a} = \vec{b} \iff a_1 = b_1, a_2 = b_2, a_3 = b_3$
• Scalar Multiplication: $\lambda \vec{a} = (\lambda a_1)\hat{i} + (\lambda a_2)\hat{j} + (\lambda a_3)\hat{k}$
• Collinearity: $\vec{a}$ and $\vec{b}$ are collinear if and only if $\vec{b} = \lambda \vec{a} \iff \frac{b_1}{a_1} = \frac{b_2}{a_2} = \frac{b_3}{a_3} = \lambda$

For points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the vector $\vec{P_1P_2}$ is given by:

Its magnitude is:

Let $P$ and $Q$ be two points with position vectors $\vec{a}$ and $\vec{b}$. Let $R$ be a point dividing the line segment $PQ$ in the ratio $m : n$:

• Internal Division: The position vector of $R$ is:$$\vec{OR} = \frac{m\vec{b} + n\vec{a}}{m + n}$$
• External Division: The position vector of $R$ is:$$\vec{OR} = \frac{m\vec{b} - n\vec{a}}{m - n}$$
• Midpoint Formula: If $R$ is the midpoint of $PQ$ ($m = n$), then:$$\vec{OR} = \frac{\vec{a} + \vec{b}}{2}$$

The scalar product of two non-zero vectors $\vec{a}$ and $\vec{b}$, denoted by $\vec{a} \cdot \vec{b}$, is defined as:

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ ($0 \le \theta \le \pi$).

• Properties:
  • $\vec{a} \cdot \vec{b}$ is a real number (scalar).
  • $\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$ (for non-zero vectors).
  • If $\theta = 0$, $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|$. Thus, $\vec{a} \cdot \vec{a} = |\vec{a}|^2$.
  • If $\theta = \pi$, $\vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}|$.
  • For mutually perpendicular unit vectors:$$\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} = 1 \quad \text{and} \quad \hat{i}\cdot\hat{j} = \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0$$
  • Angle between vectors: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$.
  • Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.
  • Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$.
  • Component Form: If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then:$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$
• Projection of a Vector on a Line:
  • Projection of vector $\vec{a}$ on vector $\vec{b}$ is given by:$$\text{Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \vec{a} \cdot \hat{b}$$
  • If $\alpha, \beta, \gamma$ are direction angles of $\vec{a}$, then its projections along $x, y, z$ axes are $a_1 = |\vec{a}|\cos\alpha$, $a_2 = |\vec{a}|\cos\beta$, $a_3 = |\vec{a}|\cos\gamma$.
• Important Inequalities:
  • Cauchy-Schwartz Inequality: $|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|$
  • Triangle Inequality: $|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$

The vector product of two non-zero vectors $\vec{a}$ and $\vec{b}$, denoted by $\vec{a} \times \vec{b}$, is defined as:

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ ($0 \le \theta \le \pi$), and $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{a}, \vec{b}, \hat{n}$ form a right-handed system.

• Properties:
  • $\vec{a} \times \vec{b}$ is a vector.
  • $\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$ (for non-zero vectors).
  • If $\theta = \frac{\pi}{2}$, $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|$.
  • For unit vectors:$$\hat{i}\times\hat{i} = \hat{j}\times\hat{j} = \hat{k}\times\hat{k} = \vec{0}$$$$\hat{i}\times\hat{j} = \hat{k}, \quad \hat{j}\times\hat{k} = \hat{i}, \quad \hat{k}\times\hat{i} = \hat{j}$$$$\hat{j}\times\hat{i} = -\hat{k}, \quad \hat{k}\times\hat{j} = -\hat{i}, \quad \hat{i}\times\hat{k} = -\hat{j}$$
  • Angle between vectors: $\sin\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}$.
  • Non-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$.
  • Distributive: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$.
  • Determinant Form: If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then:$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$
• Geometric Applications:
  • Area of a Triangle: If $\vec{a}$ and $\vec{b}$ represent adjacent sides of a triangle, then:$$\text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}|$$
  • Area of a Parallelogram: If $\vec{a}$ and $\vec{b}$ represent adjacent sides of a parallelogram, then:$$\text{Area} = |\vec{a} \times \vec{b}|$$