Vectors in Maths

Vectors play a crucial role in mathematics, especially in understanding the geometric and physical significance of various concepts. They are not just confined to one branch but span across multiple fields such as physics, engineering, and computer science. This blog delves into the world of vectors in maths, focusing on their definitions, properties, and applications in class 12 mathematics.

1.0What are Vectors in Maths?

In mathematics, a vector is defined as an entity that has both magnitude and direction. Unlike scalar quantities, which are described only by their magnitude (such as temperature, mass, or time), vectors provide a more comprehensive representation that includes both size and direction. This dual characteristic makes vectors essential in describing physical phenomena such as displacement, velocity, and acceleration.

2.0Types of Vectors

Vectors in maths can be classified into various types depending on their properties and orientation. Some of the common types include:

1. Zero Vector (Null Vector): A vector with zero magnitude and no specific direction. It is usually represented as 0 or 0 .
2. Unit Vector: A vector with a magnitude of one, usually used to denote direction. It is represented as $\hat{i},\hat{j},\text{ or }\hat{k}$ for directions along the x, y, and z axes, respectively.
3. Position Vector: A vector that represents the position of a point relative to the origin.
4. Co-planar Vectors: Vectors that lie in the same plane.
5. Equal Vectors: Vectors that have the same magnitude and direction, regardless of their initial points.

These categories help in understanding and manipulating vectors in mathematical problems, especially in vectors 12^th maths concepts.

3.0Properties of Mathematical Vectors

Mathematical vectors have several properties that make them versatile and useful:

1. Addition of Vectors: Vectors can be added using the triangle or parallelogram law. The resultant vector, in this case, represents the combined effect of the two vectors.  

• Triangle Law: If two vectors are represented by two sides of a triangle taken in order, then their sum is given by the third side of the triangle taken in the reverse order.
• Parallelogram Law: If two vectors are represented by adjacent sides of a parallelogram, their resultant is given by the diagonal of the parallelogram passing through their intersection.

2. Scalar Multiplication: A vector can be multiplied by a scalar (a real number), which changes its magnitude but not its direction (unless the scalar is negative, in which case the direction is reversed).
3. Dot Product (Scalar Product): The dot product of two vectors is a scalar value determined by multiplying their magnitudes and the cosine angle between them.  This product helps in finding the projection of one vector onto another.

Geometrically, the dot product can be interpreted as:

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

Where $|\mathbf{A}|\text{ and }|\mathbf{B}|$ are the magnitudes of the vectors, and is the angle between them.

4. Cross Product (Vector Product): The cross product of two vectors results in another vector that is perpendicular to both, with a magnitude equivalent to the product of their magnitudes and the sine of the angle formed between them. The vector product of two nonzero vectors $\vec{a}\text{ and }\vec{b}$, 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 $\vec{a}\text{ and }\vec{b}$ , 0 ≤  θ ≤ π and is a unit vector perpendicular to both $\vec{a}\text{ and }\vec{b}$, such that $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\hat{n}$ form a right handed system. i.e., the right handed system rotated from $\overrightarrow{\mathrm{a}}\text{ to }\overrightarrow{\mathrm{b}}$ moves in the direction of $\hat{n}$.

4.0Understanding the Direction of Vectors

The direction of vectors is an integral part of their definition. The direction is usually described using angles or by referring to the unit vectors along each axis. For a vector $\mathbf{A}=a\hat{i}+b\hat{j}+c\hat{k}$, the direction ratios are given by a, b, and c, while the direction cosines are expressed as $\cos \alpha ,\cos \beta ,\cos \gamma$ , where $\alpha ,\beta ,\text{ and }\gamma$ are the angles the vector makes with the x, y, and z axes, respectively.

In vectors in maths class 12, understanding the direction of vectors is crucial in solving problems related to the angle between vectors, the projection of vectors, and vector resolution.

5.0Application of Vectors in Class 12 Mathematics

In 12th maths, vectors are applied applied in a variety of topics such as:

1. Three-Dimensional Geometry: Vectors are used to represent points, lines, and planes in three-dimensional space, making it easier to calculate distances, angles, and intersections.
2. Physics Applications: Vectors are extensively used to describe physical quantities like force, torque, and momentum.
3. Engineering and Computer Graphics: In engineering, vectors are fundamental in fields like statics and dynamics, while in computer graphics, they help in creating and manipulating shapes and animations.

6.0Solved Example on Vectors

Example 1: Given two vectors $\mathbf{A}=3\hat{i}-2\hat{j}+4\hat{k}$ and $\mathbf{B}=2\hat{i}+\hat{j}-3\hat{k}$ , find the following:

1. The sum of A and B.
2. The dot product of A and B.
3. The cross product of A and B.

Solution:

1. Sum of A and B :

$\mathbf{A}+\mathbf{B}=(3\hat{i}-2\hat{j}+4\hat{k})+(2\hat{i}+\hat{j}-3\hat{k})$

$\mathbf{A}+\mathbf{B}=(3+2)\hat{i}+(-2+1)\hat{j}+(4-3)\hat{k}$

$\mathbf{A}+\mathbf{B}=5\hat{i}-\hat{j}+\hat{k}$

2. Dot Product of A and B:

$\mathbf{A}\cdot \mathbf{B}=(3)(2)+(-2)(1)+(4)(-3)$

$\mathbf{A}\cdot \mathbf{B}=6-2-12$

$\mathbf{A}\cdot \mathbf{B}=-8$

3. Cross Product of A and B:

$\mathbf{A}\times \mathbf{B}=$ $\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -2& 4\\ 2& 1& -3\end{array}\right|$

$\mathbf{A}\times \mathbf{B}=\hat{i}\{[(-2)(-3)-(4)(1)]-\hat{j}[(3)(-3)-(4)(2)]+\hat{k}[(3)(1)-(-2)(2)]$

$\mathbf{A}\times \mathbf{B}=\hat{i}(6-4)-\hat{j}(-9-8)+\hat{k}(3+4)$

$\mathbf{A}\times \mathbf{B}=2\hat{i}+17\hat{j}+7\hat{k}$

Example 2: Given a vector $\mathbf{A}=4\hat{i}-3\hat{j}+5\hat{k}$ , find its magnitude and the direction cosines.

Solution:

Magnitude of A :

The magnitude (or length) of vector A is given by:

$|\mathbf{A}|=\sqrt{(4{)}^2+(-3{)}^2+(5{)}^2}$

$|\mathbf{A}|=\sqrt{16+9+25}$

$|\mathbf{A}|=\sqrt{50}$

$|\mathbf{A}|=5\sqrt{2}$

Direction Cosines:

The direction cosines are given by $\cos \alpha ,\cos \beta \text{, and }\cos \gamma$, where $\alpha ,\beta ,\gamma$ are the angles the vector makes with the x, y, and z axes, respectively.

• $\cos \alpha =\frac{4}{|\mathbf{A}|}=\frac{4}{5\sqrt{2}}$
• $\cos \beta =\frac{-3}{|\mathbf{A}|}=\frac{-3}{5\sqrt{2}}$
• $\cos \gamma =\frac{5}{|\mathbf{A}|}=\frac{5}{5\sqrt{2}}=\frac{1}{\sqrt{2}}$

Thus, the magnitude is $5\sqrt{2}$  and the direction cosines are $\frac{4}{5\sqrt{2}},\frac{-3}{5\sqrt{2}}$, and $\frac{1}{\sqrt{2}}$ .

Example 3: Calculate the unit vector in the direction of $\mathbf{B}=-7\hat{i}+24\hat{j}+10\hat{k}$.

Solution:

The unit vector $\hat{B}$ along the direction of a vector B is given by:

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

First, find the magnitude of B :

$|\mathbf{B}|=\sqrt{(-7{)}^2+(24{)}^2+(10{)}^2}$

$|B|=\sqrt{49+576+100}$

$|\mathbf{B}|=\sqrt{725}$

$|B|=5\sqrt{29}$

Now, the unit vector is:

$\hat{B}=\frac{-7\hat{i}+24\hat{j}+10\hat{k}}{5\sqrt{29}}$

Thus, the unit vector is:

$\hat{B}=\frac{-7}{5\sqrt{29}}\hat{i}+\frac{24}{5\sqrt{29}}\hat{j}+\frac{10}{5\sqrt{29}}\hat{k}$

Example 4: Determine if the vectors $\mathbf{P}=2\hat{i}+4\hat{j}+6\hat{k}$ and $\mathbf{Q}=1\hat{i}+2\hat{j}+3\hat{k}$ are collinear.

Solution:

Two vectors are collinear if one is a scalar multiple of the other. Let's check if we can express P  as kQ.

$\mathbf{P}=2\hat{i}+4\hat{j}+6\hat{k}$

$\mathbf{Q}=1\hat{i}+2\hat{j}+3\hat{k}$

If P = kQ, then:

$2=k\cdot 1,4=k\cdot 2,6=k\cdot 3$

Solving these, we get k = 2 in all cases. Therefore, P is 2Q, implying that P and Q are collinear.

Example 5: Find the angle between vectors $\mathbf{A}=3\hat{i}+4\hat{j}$ and $\mathbf{B}=4\hat{i}-3\hat{j}$.

Solution:

The angle θ between two vectors A and B is given by the formula:

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

Find $\mathrm{A}\cdot \mathrm{B}$ :

$\mathbf{A}\cdot \mathbf{B}=(3)(4)+(4)(-3)$

$\mathbf{A}\cdot \mathbf{B}=12-12$

$\mathbf{A}\cdot \mathbf{B}=0$

Find the magnitudes of A and B:

$|\mathbf{A}|=\sqrt{(3{)}^2+(4{)}^2}=\sqrt{9+16}=\sqrt{25}=5$

$|\mathbf{B}|=\sqrt{(4{)}^2+(-3{)}^2}=\sqrt{16+9}=\sqrt{25}=5$

Calculate $\cos \theta$ :

$\cos \theta =\frac{0}{5\cdot 5}=0$

Since $\cos \theta =0$ , the angle between the vectors is 90° .

7.0Practice Questions on Vectors in Maths

1. Find the magnitude and direction cosines of the vector $\mathbf{C}=6\hat{i}-2\hat{j}+3\hat{k}$.
2. Calculate the magnitude of the resultant vector formed by the addition of $\mathbf{A}=7\hat{i}+3\hat{j}-5\hat{k}$ and $\mathbf{B}=-4\hat{i}+6\hat{j}+2\hat{k}$ .
3. Determine whether the vectors $\mathbf{P}=5\hat{i}-2\hat{j}+3\hat{k}$ and $\mathbf{Q}=10\hat{i}-4\hat{j}+6\hat{k}$ are collinear.
4. If $\mathbf{U}=8\hat{i}-12\hat{j}$ and $\mathbf{V}=k\hat{i}-18\hat{j}$ are collinear, find the value of k.
5. Determine the dot product of the vectors $\mathbf{A}=3\hat{i}-\hat{j}+2\hat{k}$ and $\mathbf{B}=4\hat{i}+2\hat{j}-\hat{k}$ .

8.0Sample Questions on Vectors in Maths

1. What do direction cosines and direction ratios of a vector represent?

Ans: Direction cosines are the cosines of the angles that a vector makes with the positive directions of the coordinate axes. If $\mathbf{A}=a\hat{i}+b\hat{j}+c\hat{k}$ , then the direction cosines are $\cos \alpha ,\cos \beta ,\text{ and }\cos \gamma$ where:

$\cos \alpha =\frac{a}{|\mathbf{A}|},\cos \beta =\frac{b}{|\mathbf{A}|},\cos \gamma =\frac{c}{|\mathbf{A}|}$

Direction ratios are any set of numbers proportional to the direction cosines.

2. How do you find the angle between two vectors?

Ans: The angle θ between two vectors A and B can be found using the formula:

$\cos \theta =\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}$where $\mathbf{A}\cdot \mathbf{B}$ is the dot product of the vectors and $|\mathbf{A}|and|\mathbf{B}|$ are their magnitudes.

3. What is the cross product of two vectors?

Ans: The cross product (or vector product) of two vectors $\mathbf{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\mathbf{B}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ is given by:

$\mathbf{A}\times \mathbf{B}=\left(a_2{b}_3-a_3{b}_2\right)\hat{i}-\left(a_1{b}_3-a_3{b}_1\right)\hat{j}+\left(a_1{b}_2-a_2{b}_1\right)\hat{k}$

The cross product is a vector that is orthogonal to A and B both.

4. How do you find the projection of one vector onto another?

Ans: The projection of vector A onto vector B is given by:

$\text{ Projection of }\mathbf{A}\text{ on }\mathbf{B}=\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{B}{|}^2}\mathbf{B}$

This formula calculates the component of A along the direction of B.

5. What are unit vectors and how are they used?

Ans: A unit vector is a vector with a magnitude of 1, typically used to indicate direction. If A is any non-zero vector, the unit vector in its direction is: $\hat{A}=\frac{\mathbf{A}}{|\mathbf{A}|}$