Answer

Verified

415.2k+ views

**Hint:**The given problem statement states to find the angle between the vectors with the help of the scalar product. You can assume each vector a variable that means one vector will be “a” and another one will be “b”. So, let’s look at the approach of the problem statement.

**Complete Complete Step by Step Solution:**

The given problem statement is to find the angle between the vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$.

Now, the first thing is to assume each of the vectors, that means, we will let $\hat{i}-2\hat{j}+3\hat{k}$is $\vec{a}$and $3\hat{i}-2\hat{j}+\hat{k}$ is $\vec{b}$.

So, if we write the vectors in a different manner, that means,

$\Rightarrow \vec{a}=1\hat{i}-2\hat{j}+3\hat{k}$

$\Rightarrow \vec{b}=3\hat{i}-2\hat{j}+1\hat{k}$

Now, we will use the scalar product formula, that means, we get,

$\Rightarrow \vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos \theta $ where ,$\theta $is the angle between $\vec{a}$ and $\vec{b}$.

Now, we will find$\vec{a}.\vec{b}$, that means, we get,

$\Rightarrow \vec{a}.\vec{b}=(1\hat{i}-2\hat{j}+3\hat{k}).(3\hat{i}-2\hat{j}+1\hat{k})$

Now, when we solve this above equation, we get,

$\Rightarrow \vec{a}.\vec{b}=[(1\hat{i}.3\hat{i})+(-2\hat{j}.-2\hat{j})+(3\hat{k}.1\hat{k})]$

We have kept$\hat{i}.\hat{i}$, $\hat{j}.\hat{j}$ and $\hat{k}.\hat{k}$ because all these have the values 1 instead of $\hat{i}.\hat{j}$, $\hat{j}.\hat{k}$ and $\hat{k}.\hat{i}$because all these have the values as 0.

$\Rightarrow \vec{a}.\vec{b}=[(1.3)+(-2.-2)+(3.1)]$

$\Rightarrow \vec{a}.\vec{b}=3+4+3$

Now, when we solve, we get,

$\Rightarrow \vec{a}.\vec{b}=10$

Now, we will find magnitude of $\vec{a}$, that means, we get,

$\Rightarrow \vec{a}=1\hat{i}-2\hat{j}+3\hat{k}$

$\Rightarrow \vec{a}=\sqrt{{{(1)}^{2}}+{{(-2)}^{2}}+{{(3)}^{2}}}$

Now, when we solve, we get,

$\Rightarrow \vec{a}=\sqrt{1+4+9}$

$\Rightarrow \vec{a}=\sqrt{14}$

Similarly, we will find magnitude of $\vec{b}$, that means, we get,

$\Rightarrow \vec{b}=3\hat{i}-2\hat{j}+1\hat{k}$

$\Rightarrow \vec{b}=\sqrt{{{(3)}^{2}}+{{(-2)}^{2}}+{{(1)}^{2}}}$

Now, when we solve, we get,

$\Rightarrow \vec{b}=\sqrt{9+4+1}$

$\Rightarrow \vec{b}=\sqrt{14}$

Now, we will put the respective values in the formula $\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos \theta $, we get,

$\Rightarrow 10=\sqrt{14}\sqrt{14}\cos \theta $

Now, we will rearrange the equation, we get,

$\Rightarrow \dfrac{10}{\sqrt{14}\sqrt{14}}=\cos \theta $

As, we now $\sqrt{y}\sqrt{y}=y$, similarly we will apply in the equation and then we will convert it to lowest terms, we get,

$\Rightarrow \dfrac{10}{14}=\cos \theta $

$\Rightarrow \dfrac{5}{7}=\cos \theta $

Now when we rearrange we will get the value of $\theta $, we get,

$\Rightarrow {{\cos }^{-1}}(\dfrac{5}{7})=\theta $

After rearranging the equation, we get,

$\Rightarrow \theta ={{\cos }^{-1}}(\dfrac{5}{7})$

**Therefore, the value of $\theta $or the angle between two vectors is ${{\cos }^{-1}}(\dfrac{5}{7})$.**

**Note:**

In the above problem statement, we have used the fine concept of the scalar product. The scalar product is also known as the dot product. The formula used in the scalar product is $\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos \theta $. You need to note that the scalar product is commutative as well as distributive.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE