Hello friends, today I will talk about direction cosines of vectors. What are these things? Also, some solved examples of direction cosines of vectors.

## Direction cosines of vectors

#### What are direction cosines of vectors?

‘Direction cosines’ are the cosines of angles. These angles are made by the given vector with axes of reference. Here axes of reference mean x-axis, y-axis and z-axis.

Now I will give you some examples.

#### Examples of direction cosines of vectors

Disclaimer: 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 modulus and direction cosine of each of the vectors and Find also the modulus and the direction cosines of their sum.”

##### Solution

Here the given vectors are and

First of all, I will give them some name.

###### Step 1

Let

Now modulus of vector is

So this gives

In the same way, I will also get the modulus of vectors and .

Thus the modulus of vector is

Now I’ll simplify it to get

Similarly, the modulus of vector will be

Thus it becomes

###### Step 2

Now I will get the direction cosines of these vectors.

As per the formula, if any vector is the direction cosines of vector are

Therefore, I can say that the direction cosines of vector are

In the same way, the direction cosines of vector are

Similarly, the direction cosines of vector are

Next, I will solve the second part of the problem.

###### Step 3

The sum of the three vectors and are

Now I’ll simplify it to get

Thus the modulus of the sum of these three vectors will be

Therefore direction cosines of vector will be

Hence I can say that this is the end of this example.

Now I will go to the second example.

##### Example 2

According to Stroud and Booth (2013)* “If the position vectors of and are and respectively, find and determine its direction cosines.”

##### Solution

Here the given vectors are and

Therefore the vector will be

which gives

So this means

Hence the modulus of the vector will be

Now I’ll simplify it to get

As a result, the direction cosines of the vector are

Therefore I can conclude that the vector is . Also it’s direction cosines are

This is the answer to the given example.

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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