Question

Assertion: Angle between $\hat{i}+\hat{j}$ and $\hat{i}$ is $45{}^\circ $.
Reason: $\hat{i}+\hat{j}$ is equally inclined to both $\hat{i}$ and $\hat{j}$ and the angle between $\hat{i}$ and $\hat{j}$ and the angle between $\hat{i}$ and $\hat{j}$ is $90{}^\circ $.
A. Both assertion and reason are correct and reason is the correct explanation for assertion
B. Both assertion and reason are correct but reason is not the correct explanation for assertion.
C. Assertion is correct but reason is incorrect.
D. Both assertion and reason are incorrect.

Answer 1

Hint:In this question, we used the formula of the dot product of two vectors. Then by putting the given vectors in the formula, we will be able to find the cosine angle between the two vectors and with the help of that we will be able to choose the correct option.

Formula used :-
We will use the following formula :-
$\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos \theta $
Here, $\theta$ is the angle between vectors a and b.

Complete step by step solution:
We know the formula of dot product as
$\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos \theta $
Given vectors are
$\vec{a}=\hat{i}+\hat{j}$
$\Rightarrow \vec{b}=\hat{i}$
This can be also written as,
$\cos \theta =\dfrac{1}{\sqrt{1+1}}$

Now we substitute the vectors in the above equation, we get
$\cos \theta =\dfrac{(\hat{i}+\hat{j})(\hat{i})}{|\hat{i}+\hat{j}||\hat{i}|}$
$\Rightarrow \cos \theta =\dfrac{1}{\sqrt{1+1}}$
Then,
$\cos \theta =\dfrac{1}{\sqrt{2}}$
$\therefore \theta =45{}^\circ $
Hence both assertion and reason are correct and reason is the correct explanation of assertion.

Hence, option A is the correct answer.

Note: A scalar is that quantity which has only magnitude but in vector we have both magnitude and direction. A line of a given length and pointing along a given direction, like an arrow is a typical representation of a vector. While comparing two vector quantities of the same type, you have to compare both the magnitude and the direction and for scalars, you have to compare only magnitude.