Vector Algebra

Vector Algebra is a branch of mathematics that deals with quantities having both magnitude and direction. Unlike simple numbers, vectors can represent more complex entities such as force, velocity, and displacement.

In Vector Algebra, we use arrows to visually represent vectors. The length of the arrow indicates the vector's magnitude, while the direction of the arrow shows its direction. This graphical representation helps in understanding and solving problems in physics, engineering, and computer science.

Key operations in Vector Algebra include addition, subtraction, and multiplication of vectors. These operations follow specific rules, such as the Triangle Law and Parallelogram Law, which help in combining vectors and analyzing their interactions.

In this article, we will explore the essential concepts of Vector Algebra, including the definition and properties of vectors, key operations such as addition, subtraction, and multiplication, and the practical applications of vectors in fields like physics, engineering, and computer graphics. By the end, you'll have a solid understanding of how vectors are used to model and solve real-world problems efficiently.

1.0What is Vector Algebra

Vector Algebra is a branch of mathematics that deals with vectors, which are mathematical objects that have both magnitude (size) and direction. In vector algebra, vectors are represented as directed line segments or arrows in space. They are used to represent quantities such as force, velocity, acceleration, and displacement in physics, as well as other concepts in various fields like engineering, computer graphics, and economics.

Vector algebra involves operations such as addition, subtraction, and multiplication of vectors, which follow specific rules and properties. These operations allow for the manipulation and analysis of vectors to solve problems and describe physical phenomena. Key concepts in vector algebra include vector components, dot product, cross product, vector spaces, and applications in geometry, mechanics, and other areas of mathematics and science.

2.0Types of Vectors in Vector Algebra

In vector algebra, vectors can be classified into various types based on their properties and roles in mathematical operations. Here are the main types of vectors:

1. Zero Vector (Null Vector)

Definition: A vector with a magnitude of zero and no specific direction.

Notation: 0

Example: $0=0\hat{\mathbf{i}}+0\hat{\mathbf{j}}+0\hat{\mathbf{k}}$

2. Unit Vector

Definition: A vector with a magnitude of one, used to indicate direction.

Notation: $\hat{\mathrm{A}}$

Example: If $\mathrm{A}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$, then $\hat{\mathbf{A}}=\frac{\mathbf{A}}{|\mathbf{A}|}=\frac{3\mathbf{i}+4\mathbf{j}}{5}$

3. Position Vector

Definition: A vector that represents the position of a point relative to the origin.

Notation: r

Example: The position vector of a point P (x, y, z) is $r=x\hat{i}+y\hat{j}+z\hat{k}$

4. Co-initial Vectors

Definition: Vectors that have the same initial point.

Example: Vectors A and B starting from the same point O.

5. Collinear Vectors

Definition: Vectors that lie along the same line or are parallel to each other.

Example: Vectors $A=2\hat{i}+3\hat{j}$

and $B=4\hat{i}+6\hat{j}$ are collinear.

6. Equal Vectors

Definition: Vectors that have the same magnitude and direction, regardless of their initial points.

Example: $\mathrm{A}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ and $B=3\hat{i}+4\hat{j}$

7. Negative of a Vector

Definition: A vector that has the same magnitude as a given vector but points in the opposite direction.

Example: If $A=5\hat{i}-2\hat{j}$, then $-A=-5\hat{i}+2\hat{j}$

8. Parallel Vectors

Definition: Vectors that have the same or opposite direction.

Example: Vectors A and KA (where k is a scalar) are parallel.

9. Orthogonal (Perpendicular) Vectors

Definition: Vectors that are at right angles (90 degrees) to each other.

Example: If $A=\hat{i}+2\hat{j}$ and $B=-2\hat{i}+\hat{j}$, then A. B = 0 

10. Coplanar Vectors

Definition: Vectors that lie in the same plane.

Example: Vectors $A=2\hat{i}+3\hat{j}$ and B=$-\hat{i}+4\hat{j}$ are coplanar with any vector in the xy-plane.

These different types of vectors help in classifying and understanding their properties and how they interact with each other in various mathematical and physical contexts.

3.0Vector Algebra Operations

Vector Algebra Operations encompass a variety of mathematical manipulations involving vectors. These operations are fundamental in understanding and analyzing vectors in various fields such as physics, engineering, and computer science. Here are the primary Vector Algebra Operations:

Vector Addition

Let $\overrightarrow{\mathrm{a}}$  and $\overrightarrow{\mathrm{b}}$ be two vectors in a plane, which are represented by $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{CD}}$. Their addition can be performed in the following two ways.

Triangle Law of Addition of Vectors

If two vectors are represented in magnitude & direction by two sides of a triangle taken in same order, then their sum is represented by the third side taken in reverse order.

Parallelogram Law of Addition of Vectors

If two vectors be represented in magnitude and direction by the two adjacent sides of a parallelogram then their sum will be represented by the diagonal through the co-initial point.

Vector Subtraction

Vector $-\vec{b}$ has length equals to vector $\vec{b}$ but its direction is opposite. Subtraction of vector $\vec{a}$ and $\vec{b}$ is defined as addition of $\vec{a}$ and ($-\vec{b}$). It is written as follows:

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

Scalar Multiplication or Scalar Product

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors inclined at an angle θ. Then the scalar product of $\vec{a}$ with $\vec{b}$ is denoted by $\vec{a}\cdot \vec{b}$ and is defined as $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta ;0\le \theta \le \pi$ .

1. $\vec{a}\cdot \vec{b}=|\vec{a}\Vert \vec{b}|\cos \theta (0\le \theta \le \pi )$

Note: if θ is acute then $\vec{a}\cdot \vec{b}>0$ & if θ is obtuse then $\vec{a}\cdot \vec{b}<0$

2. (i) $\vec{a}\cdot \vec{a}=|\vec{a}{|}^2={\vec{a}}^2$

(ii) $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}(commutative)$

3. $\vec{a}\cdot (\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$ (distributive)
4. $\vec{a}\cdot \vec{b}=0\iff \vec{a}\perp \vec{b};(\vec{a},\vec{b}\ne \overrightarrow{0})$
5. $\hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1;\hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$

Linear Combination

A vector $\vec{r}$ is said to Linear Combination of vectors $\vec{a},\vec{b},\vec{c},\dots$ if ∃ scalars x, y, z such that

$\vec{r}=\mathrm{x}\vec{a}+\mathrm{y}\vec{b}+\mathrm{z}\vec{c}$

Dot Product

If $\vec{a}=a_1\hat{i}+2a\hat{j}+a_3\hat{k}\&=\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ then $\vec{a}\cdot \vec{b}$= a₁b₁ + a₂b₂+ a₃b₃ &

$\cos \theta =\frac{\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}=\frac{{\mathrm{a}}_1{\mathrm{b}}_1+{\mathrm{a}}_2{\mathrm{b}}_2+{\mathrm{a}}_3{\mathrm{b}}_3}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{a}}_2^2+{\mathrm{a}}_3^2}\cdot \sqrt{{\mathrm{b}}_1^2+{\mathrm{b}}_2^2+{\mathrm{b}}_3^2}}$

Cross Product

The vector product of two nonzero vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ , is denoted by  $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$ and defined as $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}\Vert \overrightarrow{\mathrm{b}}|\sin \theta \hat{\mathrm{n}}$

Where, θ is the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ , 0 ≤  θ ≤ π and $\hat{n}$ is a unit vector perpendicular to both $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, such that $\vec{a}$, $\vec{b}$ and $\hat{\mathbf{n}}$ form a right handed system. i.e., the right handed system rotated from $\overrightarrow{\mathrm{a}}$ to $\vec{b}$ moves in the direction of $\hat{n}$.

If either $\overrightarrow{\mathrm{a}}=\overrightarrow{0}or\overrightarrow{\mathrm{b}}=\overrightarrow{0}$, then θ is not defined and in this case , we define $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\overrightarrow{0}$.

4.0Vector Algebra Formulas

1. Vector Addition

$\mathbf{A}+\mathbf{B}=\left({\mathrm{A}}_{\mathrm{x}}+{\mathrm{B}}_{\mathrm{x}}\right)\hat{\mathbf{i}}+\left({\mathrm{A}}_{\mathrm{y}}+{\mathrm{B}}_{\mathrm{y}}\right)\hat{\mathbf{j}}+\left({\mathrm{A}}_{\mathrm{z}}+{\mathrm{B}}_{\mathrm{z}}\right)\hat{\mathbf{k}}$

Combine corresponding components of vectors A and B.

2. Vector Subtraction

$\mathbf{A}-\mathbf{B}=\left({\mathrm{A}}_{\mathrm{x}}-{\mathrm{B}}_{\mathrm{x}}\right)\hat{\mathbf{i}}+\left({\mathrm{A}}_{\mathrm{y}}-{\mathrm{B}}_{\mathrm{y}}\right)\hat{\mathbf{j}}+\left({\mathrm{A}}_{\mathrm{z}}-{\mathrm{B}}_{\mathrm{z}}\right)\hat{\mathbf{k}}$

Subtract corresponding components of vector B and A.

3. Scalar Multiplication

$k\mathbf{A}=\left(k{A}_x\right)\hat{\mathbf{i}}+\left(k{A}_y\right)\hat{\mathbf{j}}+\left(k{A}_z\right)\hat{\mathbf{k}}$

Multiply each component of vector A by the scalar k.

4. Dot Product

$\mathbf{A}\cdot \mathbf{B}={\mathrm{A}}_{\mathrm{x}}{\mathrm{B}}_{\mathrm{x}}+{\mathrm{A}}_{\mathrm{y}}{\mathrm{B}}_{\mathrm{y}}+{\mathrm{A}}_{\mathrm{z}}{\mathrm{B}}_{\mathrm{z}}$

Calculate the sum of the products of the corresponding components of vectors A and B.

Alternatively, using magnitudes and the cosine of the angle  θ between the vectors:

$\mathbf{A}\cdot \mathbf{B}=|\mathbf{A}\Vert \mathbf{B}|\cos \theta$

Note: If $\vec{A}$ and $\vec{B}$ are Parallel then $\vec{A}\times \vec{B}$ are Perpendicular then $\vec{A}\cdot \vec{B}=\overrightarrow{0}$

5. Cross Product

$\mathbf{A}\times \mathbf{B}=\left(A_y{B}_z-A_z{B}_y\right)\hat{\mathbf{i}}-\left(A_x{B}_z-A_z{B}_x\right)\hat{\mathbf{j}}+\left(A_x{B}_y-A_y{B}_x\right)\hat{\mathbf{k}}$

Compute a vector perpendicular to both A and B with magnitude equal to the area of the parallelogram they span.

6. Vector Magnitude

$|\mathbf{A}|=\sqrt{{\mathrm{A}}_{\mathrm{x}}^2+{\mathrm{A}}_{\mathrm{y}}^2+{\mathrm{A}}_{\mathrm{z}}^2}$

Calculate the length of vector A using its components.

7. Unit Vector

$\hat{\mathbf{A}}=\frac{\mathbf{A}}{|\mathbf{A}|}$

Divide vector A by its magnitude to get a unit vector in the same direction.

8. Projection of A onto B

${\mathrm{proj}}_{\mathbf{B}}\mathbf{A}=\left(\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{B}{|}^2}\right)\mathbf{B}$

Project vector A onto B, resulting in a vector parallel to B.

9. Angle Between Two Vectors

$\cos \theta =\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{A}\Vert \mathbf{B}|}$

Find the cosine of the angle between vectors A and B.

5.0Vector Algebra Examples

Example 1: Find unit vector of $\overrightarrow{\mathrm{A}}=3\hat{i}+4\hat{j}-5\hat{k}$ 

Solution:

$|\vec{A}|=\sqrt{9+16+25}=\sqrt{50}$ ⇒ unit vector: $\frac{3\hat{i}+4\hat{j}-5\hat{k}}{\sqrt{50}}$

Example 2: If $A\equiv (2\hat{i}+3\hat{j}),B\equiv (p\hat{i}+9\hat{j})$ and $C\equiv (\hat{\mathrm{i}}-\hat{\mathrm{j}})$ are collinear, then the value of p is:

(A) 1/2 (B) 3/2 (C) 7/2 (D) 5/2

Ans. (C)

Solution:

$\overrightarrow{\mathrm{AB}}=(\mathrm{p}-2)\hat{\mathrm{i}}+6\hat{\mathrm{j}},\overrightarrow{\mathrm{AC}}=-\hat{\mathrm{i}}-4\hat{\mathrm{j}}$

Now A, B, C are collinear ⇔  $\overrightarrow{AB}\Vert \overrightarrow{AC}$ ⇔ $\frac{p-2}{-1}=\frac{6}{-4}$  ⇔ $\mathrm{p}=\frac{7}{2}$

Example 3: The value of λ when $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{b}=8\hat{i}+\lambda \hat{j}+4\hat{k}$ are parallel is:

(A) 4 (B) –6 (C) –12 (D) 1

Ans. (C)

Solution:

Since $\vec{a}$ & $\vec{b}$ are parallel ⇒$\frac{2}{8}$=$-\frac{3}{\lambda }$=$\frac{1}{4}$ ⇒  –12

Example 4: If $\vec{a}$ and $\vec{b}$ are unit vectors, then which of the following values of $\vec{a}\cdot \vec{b}$ is not possible?

(A) $\sqrt{3}$ (B) $\frac{\sqrt{3}}{2}$

(C) $\frac{1}{\sqrt{2}}$ (D) $\frac{-1}{2}$

Ans. (A)

Solution:

Since $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta =\cos \theta$ and cosθ never be equal to $\sqrt{3}$

Example 5: If p^th, q^th, r^th terms of a G.P. are the positive numbers a, b, c then angle between the vectors $\log a^2\hat{i}+\log b^2\hat{j}+\log c^2\hat{k}$ and (q-r) $\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$ is: 

(A) $\frac{\mathrm{p}}{3}$

(B) $\frac{p}{2}$

(C) ${\sin }^{-1}\left(\frac{1}{\sqrt{a^2+b^2+c^2}}\right)$

(D) none of these

Ans. (B)

Solution:

Let x₀ be first term and x the common ratio of the G.P.

∴ a = x₀ x^p-1  , b = x₀x^q–1, c = x₀x^r-1  

⇒  log a = log  x₀ + (p–1)  log x  ; log b = log x₀ +(q–1)  log x;  log c = log x₀ +(r–1) log x

If  $\vec{a}=\log a^2\hat{i}+\log b^2\hat{j}+\log c^2\hat{k}$ and $\vec{b}=(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$

$\therefore \vec{a}\cdot \vec{b}=\Sigma 2(\log a)(q-r)=2\sum \left(\log x_0+(p-1)\log x\right)(q-r)=0$

 ⇒${\vec{a}}^{\wedge }\vec{b}=\frac{\pi }{2}.$ 

Example 6: A unit vector perpendicular to the plane determined by the points (1,–1,2), (2,0,–1) and (0,2,1) is 

(A) $\pm \frac{1}{\sqrt{6}}(2\hat{i}+\hat{j}+\hat{k})$

(B) $\frac{1}{\sqrt{6}}(2\hat{i}+\hat{j}+\hat{k})$

(C) $\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+\hat{k})$

(D) $\frac{1}{\sqrt{6}}(2\hat{i}-\hat{j}-\hat{k})$

Ans. (A)

Solution:

$\vec{a}=\hat{i}+\hat{j}-3\hat{k},\vec{b}=-2\hat{i}+2\hat{j}+2\hat{k}$

$\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -3\\ -2& 2& 2\end{array}\right|=8\hat{i}+4\hat{j}+4\hat{k}$

Hence unit vector = $\pm \frac{2\hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}$

Example 7: If $\vec{a}=2\hat{i}+3\hat{j}-5\hat{k},\vec{b}=m\hat{i}+n\hat{j}+12\hat{k}$ and $\vec{a}\times \vec{b}=0$ then (m,n)=

(A) $\left(-\frac{24}{5},\frac{36}{5}\right)$ (B) $\left(\frac{24}{5},-\frac{36}{5}\right)$

(C) $\left(-\frac{24}{5},-\frac{36}{5}\right)$ (D) $\left(\frac{24}{5},\frac{36}{5}\right)$

Ans. (C)

Solution:

$\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -5\\ m& n& 12\end{array}\right|$=$(36+5n)\hat{i}-(24+5m)\hat{j}+(2n-3m)\hat{k}$=0

$\Rightarrow m=\frac{-24}{5},n=\frac{-36}{5}$

6.0Solved Questions on Vector Algebra

1. What is a unit vector?

Ans: A unit vector is a vector with a magnitude of one. It is used to indicate direction. If A is any vector, its unit vector is $\hat{\mathbf{A}}=\frac{\mathbf{A}}{|\mathbf{A}|}$, where $|\mathbf{A}|$ is the magnitude of A.

2. How do you find the magnitude of a vector?

Ans: The magnitude of a vector $\mathbf{A}=A_x\hat{\mathbf{i}}+A_y\hat{\mathbf{j}}+A_z\hat{\mathbf{k}}$ is calculated as $|\mathbf{A}|=\sqrt{A_x^2+A_y^2+A_z^2}$

3. What is the projection of one vector onto another?

Ans:  The projection of vector A onto vector B is given by ${\mathrm{proj}}_{\mathbf{B}}\mathbf{A}=\left(\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{B}{|}^2}\right)\mathbf{B}$. It represents the component of A in the direction of B.

5. What is a position vector?

Ans: A position vector is a vector that represents the position of a point in space relative to an origin. For a point P(x, y, z), the position vector is $\mathbf{r}=\mathrm{x}\hat{\mathbf{i}}+\mathrm{y}\hat{\mathbf{j}}+\mathrm{z}\hat{\mathbf{k}}$.

6. How do you subtract one vector from another?

Ans: To subtract vector B from vector A, you subtract their corresponding components:

$\mathbf{A}-\mathbf{B}=\left(A_x-B_x\right)\hat{\mathbf{i}}+\left(A_y-B_y\right)\hat{\mathbf{j}}+\left(A_z-B_z\right)\hat{\mathbf{k}}$