Hello friends, today I will talk about the angle between two vectors.

## Angle between two vectors

Suppose and are two vectors. Also the direction cosines of vector are and those of vector are

If is the angle between two vectors and the cosine of this angle will be

Now I will solve some problems for you.

#### Examples on the *angle between two vectors*

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2013)* “Find the cosine of the angle between the vectors and ”

##### Solution

Here the given vectors are and

Let me give them some name first.

###### Step 1

Let and

Therefore the modulus of the vector is

Thus the direction cosines of the vector are

Similarly the modulus of the vector is

Thus the direction cosines of the vector are

Let is the angle between the vectors and .

Therefore as per the formula of angle between two vectors,

Hence I can conclude that the cosine of the angle between the vectors and is – 0.4768. This is the answer to the given example.

Now I will go to the second problem.

##### Example 2

According to Stroud and Booth (2013)* “Find the direction cosines of the vectors whose direction ratios are and Hence find the angle between the two vectors.”

##### Solution

Here the direction ratios of two vectors are and

First of all, I will give them some name.

Let is the vector with direction ratio . Similarly, let is the vector with direction ratio .

Now I’ll get the direction cosines of these two vectors.

###### Step 1

I’ll start with vector .

The modulus of vector is

Thus as per the formula for direction cosines of any vector, direction cosines of vector are

Next I’ll get the direction cosines of vector .

The modulus of vector is

Therefore, direction cosines of vector are

Now I’ll do the second part of the problem.

###### Step 2

If is the angle between the vectors and , then I can say

Thus the angle will be

Hence I can conclude that the direction cosines of the vectors whose direction ratios are and are and respectively. Also the angle between these two vectors is 98.6965 degrees.

These are the answers to this problem.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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