51  Vector Algebra

51.1 Introduction

Vector Algebra is the study of quantities that have both magnitude and direction.
Unlike scalars (which have only magnitude, e.g., mass, time, temperature), vectors represent physical and geometric quantities such as displacement, velocity, acceleration, and force.

In aptitude and higher mathematics, vector algebra is used to represent geometry (lines, planes), mechanics, and space analysis.

51.2 Scalars vs Vectors

  • Scalars: described by magnitude only (speed = 60 km/h).
  • Vectors: described by both magnitude and direction (velocity = 60 km/h north).

Notation: vectors are written as bold letters (\(\vec{a}\)) or with an arrow (\(\overrightarrow{AB}\)).

51.3 Basics of Vectors

  • Magnitude: \(|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\).
  • Direction: specified by angles or direction cosines.
  • Unit vector: vector of length 1 in same direction: \(\hat{a}=\vec{a}/|\vec{a}|\).
  • Zero vector: \(\vec{0}=(0,0,0)\).
  • Position vector: vector from origin \(O(0,0,0)\) to point \(P(x,y,z)\) is \(\vec{OP}=(x,y,z)\).

51.4 Operations on Vectors

51.4.1 Addition and Subtraction

If \(\vec{a}=(a_1,a_2,a_3)\), \(\vec{b}=(b_1,b_2,b_3)\):
- \(\vec{a}+\vec{b}=(a_1+b_1,\;a_2+b_2,\;a_3+b_3)\).
- \(\vec{a}-\vec{b}=(a_1-b_1,\;a_2-b_2,\;a_3-b_3)\).

Geometric rule: parallelogram law.

51.4.2 Scalar Multiplication

\(c\vec{a}=(ca_1,ca_2,ca_3)\).
- If \(c>0\): same direction.
- If \(c<0\): opposite direction.
- If \(c=0\): zero vector.

51.4.3 Dot Product (Scalar Product)

For \(\vec{a},\vec{b}\) with angle \(\theta\):
\[ \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta \]

Component form:
\[ \vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3 \]

Properties: - \(\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}\)
- \(\vec{a}\cdot\vec{a}=|\vec{a}|^2\)
- \(\vec{a}\cdot\vec{b}=0 \iff \vec{a}\perp\vec{b}\)

51.4.4 Cross Product (Vector Product)

For \(\vec{a},\vec{b}\) with angle \(\theta\):
\[ \vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\;\hat{n} \] where \(\hat{n}\) = unit vector perpendicular to both (right-hand rule).

Determinant form:
\[ \vec{a}\times\vec{b}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix} \]

Properties: - \(\vec{a}\times\vec{b}=-(\vec{b}\times\vec{a})\).
- \(\vec{a}\times\vec{a}=\vec{0}\).
- \(|\vec{a}\times\vec{b}|\) = area of parallelogram formed by \(\vec{a},\vec{b}\).

51.4.5 Scalar Triple Product

For \(\vec{a},\vec{b},\vec{c}\): \[ \vec{a}\cdot(\vec{b}\times\vec{c})= \begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{vmatrix} \]

Interpretation: volume of parallelepiped formed by \(\vec{a},\vec{b},\vec{c}\).

51.4.6 Vector Triple Product

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

51.5 Applications in Geometry

  1. Collinearity of points: \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are parallel \(\Rightarrow \overrightarrow{AB}\times\overrightarrow{AC}=\vec{0}\).
  2. Coplanarity: \(\vec{a},\vec{b},\vec{c}\) are coplanar if \(\vec{a}\cdot(\vec{b}\times\vec{c})=0\).
  3. Equation of line in vector form:
    \[ \vec{r}=\vec{a}+\lambda\vec{b} \] (point \(\vec{a}\), direction \(\vec{b}\)).
  4. Equation of plane:
    \[ \vec{n}\cdot(\vec{r}-\vec{a})=0 \] (normal \(\vec{n}\), point \(\vec{a}\)).

51.6 Solved Examples

Example 1
Find angle between vectors \(\vec{a}=(1,2,3)\) and \(\vec{b}=(4,-5,6)\).
\(\vec{a}\cdot\vec{b}=1\cdot4+2(-5)+3\cdot6=4-10+18=12\).
\(|\vec{a}|=\sqrt{1+4+9}=\sqrt{14}\), \(|\vec{b}|=\sqrt{16+25+36}=\sqrt{77}\).
\(\cos\theta=12/(\sqrt{14}\cdot\sqrt{77})=12/\sqrt{1078}\).

Example 2
Find area of parallelogram with diagonals represented by vectors \((3,2,1)\) and \((1,-2,4)\).
Area = \(\tfrac{1}{2}|\vec{d_1}\times\vec{d_2}|\).
Compute determinant to find cross product.

Example 3
Check if vectors \((1,2,3),(4,5,6),(7,8,9)\) are coplanar.
Compute scalar triple product:
\[ \begin{vmatrix} 1&2&3\\ 4&5&6\\ 7&8&9 \end{vmatrix}=0 \] Hence coplanar.

Example 4
Equation of line through point \((1,2,3)\) parallel to vector \((2,-1,4)\).
\(\vec{r}=(1,2,3)+\lambda(2,-1,4)\).

Example 5
Equation of plane through \((2,1,-1)\) with normal \((1,-2,1)\).
\(1(x-2)-2(y-1)+1(z+1)=0 \Rightarrow x-2-2y+2+z+1=0 \Rightarrow x-2y+z+1=0\).

51.7 Practice Problems

  1. Find unit vector in direction of \((3,4)\).
  2. If \(\vec{a}=(1,2,-1)\), \(\vec{b}=(2,1,0)\), find \(\vec{a}\cdot\vec{b}\).
  3. Find \(\vec{a}\times\vec{b}\) for \(\vec{a}=(1,0,0)\), \(\vec{b}=(0,1,0)\).
  4. Find volume of parallelepiped formed by \((1,0,0),(0,1,0),(0,0,1)\).
  5. Write vector equation of line passing through \((0,1,2)\) parallel to \((1,-2,1)\).
  6. Find equation of plane passing through \((1,1,1)\) perpendicular to \((2,3,4)\).
  7. Find angle between diagonals of cube of side \(a\).

51.8 Summary

  • Vectors = magnitude + direction.
  • Dot product → projection/angle; cross product → perpendicular vector/area.
  • Scalar triple product → volume; vector triple product → expansion identity.
  • Applications: equations of line/plane, collinearity, coplanarity.
  • Vectors unify algebra and geometry, simplifying 2D and 3D problems.