In Mathematics, the topic vector algebra for class 12 is all about in-depth studying of scalars and vectors. A lot of quantities have magnitude as well as direction.

A quantity is known as a vector quantity if they have a direction as well as magnitude. A quantity is known as scalar quantity if it has only magnitude. Some examples of the vector quantities are acceleration, velocity, displacement, force, momentum, weight, and more. In this article, you will be studying vector algebra for class 12. It covers concepts such as position vector, types of vectors, the operation that is performed on vectors, and some solved examples to understand the concept better.

The physical quantities having direction and magnitude is described using the concept of vectors. A scalar quantity just has a number or has only a constant. Important concepts in vector algebra for class 12 NCERT syllabus are vector definition, position vector, types of vectors, the operation that is performed on vectors, solved examples, multiplication of a vector with a scalar quantity, and multiplication of two vectors.

Position Vector

With respect to the origin O ( 0, 0, 0 ), the position vector is given, where the coordinate of point K is given as K ( x, y, z). The formula goes by:

\[\overline{OK} = \sqrt{x^{2} + y^{2} + z^{2}}\]

The direction ratios ( l, m, n ), the magnitude ( s ), and the direction cosines ( u, v, w ) of any vector that is related are given as u = \[\frac{l}{s}\], v = \[\frac{m}{s}\], n = \[\frac{w}{ s}\].

The different types of vectors in vector algebra are:

Zero Vector

Unit Vector

Coinitial Vector

Collinear Vector

Equal Vector

Negative of a Vector

The operations that can be performed on two vectors are addition, multiplication, and subtraction. Now, let’s understand how these operations are applied to the vectors.

The dot product of the vector is calculated with the help of a dot. It is represented as shown below:

j . k

Now, the dot product can be represented in two different ways:

(image will be uploaded soon)

l . m = | l | x | m | x cos Ө

Here,

| l | - is the length (the magnitude) of vector l.

| m | - is the length (the magnitude) of vector m.

Ө - the angle between l and m.

To find the dot product of vector l and vector m, we multiply the length of l = | l | and the length of m = | m |

The other way to calculate the dot product is:

(image will be uploaded soon)

l . m = lx + mx + ly + my

First, the x values are multiplied with each other. Next, the y values are multiplied, and then they are added together.

Consider two vectors: PQ and QR. Here, the ending point of the first vector is the starting point of the second vector. Hence, the sum of two vectors is represented as:

\[\overline{PR} = \overline{PQ} + \overline{QR}\]

This is also known as the triangle law of addition of vectors and is used when one needs to find the sum of two or more vectors.

Question 1: The direction of a vector is \[\overline{v} = 4\widehat{i} + 5\widehat{j} + 6\widehat{k}\]. Find the unit vector.

Solution:

The unit vector in the direction of a vector \[\overline{v}\] is known as \[\widehat{v} = \frac{1}{|a|}\times \overline{v}\]

Hence, \[|\overline{v}| = \sqrt{4^{2} + 5^{2} + 6^{2}} = \sqrt{77}\]

Therefore, \[\widehat{v} = \frac{1}{\sqrt{77}} (4\widehat{i} + 5\widehat{j} + 6\widehat{k})\]

\[\widehat{v} = \frac{4}{\sqrt{77}}\widehat{i} + \frac{5}{\sqrt{77}}\widehat{j} + \frac{6}{\sqrt{77}}\widehat{k}\]

Question 2: The direction of a vector is \[\overline{v} = 1\widehat{i} + 7\widehat{j} + 4\widehat{k}\]. Find the unit vector.

Solution:

The unit vector in the direction of a vector \[\overline{v}\] is known as \[\widehat{v} = \frac{1}{|a|}\times \overline{v}\]

Hence, \[|\overline{v}| = \sqrt{1^{2} + 7^{2} + 4^{2}} = \sqrt{66}\]

Therefore, \[\widehat{v} = \frac{1}{\sqrt{66}} (1\widehat{i} + 7\widehat{j} + 4\widehat{k})\]

\[\widehat{v} = \frac{1}{\sqrt{66}}\widehat{i} + \frac{7}{\sqrt{66}}\widehat{j} + \frac{4}{\sqrt{66}}\widehat{k}\]

Question 3: The direction of a vector is \[\overline{v} = 3\widehat{i} + 2\widehat{j} + 3\widehat{k}\]. Find the unit vector.

Solution:

The unit vector in the direction of a vector \[\overline{v}\] is known as \[\overline{v}\] is known as \[\widehat{v} = \frac{1}{|a|}\times \overline{v}\]

Hence, \[|\overline{v}| = \sqrt{3^{2} + 2^{2} + 3^{2}} = \sqrt{22}\]

Therefore, \[\widehat{v} = \frac{1}{\sqrt{22}} (3\widehat{i} + 2\widehat{j} + 3\widehat{k})\]

\[\widehat{v} = \frac{3}{\sqrt{22}}\widehat{i} + \frac{2}{\sqrt{22}}\widehat{j} + \frac{3}{\sqrt{22}}\widehat{k}\]

FAQ (Frequently Asked Questions)

1) How to Calculate the Dot Product of Two Vectors?

The dot product of the vector is calculated with the help of a dot. It is represented as shown below j. k

Now, the dot product can be represented in two different ways:

l. m = | l | x | m | x cos Ө

Here,

| l | - is the length (the magnitude) of vector l.

| m | - is the length (the magnitude) of vector m.

Ө - the angle between l and m.

To find the dot product of vector l and vector m, we multiply the length of l = | l | and the length of m = | m |

The other way to calculate the dot product is:

l. m = l_{x} + m_{x} + l_{y} + m_{y}

First, the x values are multiplied with each other. Next, the y values are multiplied and then they are added together.