Question

Calculate the values of \(a\) and \(b\) if vectors \(a\hat{i}+b\hat{j} = \hat{n}\) and \((\hat{i}+\hat{j})\) are perpendicular to each other

• A. \(\frac{1}{2},\frac{1}{2}\)
• B. \(0,0\)
• C. \(\sqrt{2},-\frac{1}{\sqrt{2}}\)
• D. \((\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}})\)

Hint:

Perpendicular means zero dot product; unit vector means unit magnitude.

Answer 1

The Correct Option is D

Step 1: Plan:
Rotate $\hat i+\hat j$ by $90^{\circ}$ and make it a unit vector.

Step 2: Steps:
A vector perpendicular to $\hat i+\hat j$ is $\hat i - \hat j$. Its length is $\sqrt2$, so the unit vector is $\frac{\hat i - \hat j}{\sqrt2}$.
So $a = \frac{1}{\sqrt2}$ and $b = -\frac{1}{\sqrt2}$. The opposite vector (both signs flipped) would also work, but it is not among the options.

Final Answer:
The values are $a=\frac{1}{\sqrt2}$ and $b=-\frac{1}{\sqrt2}$, option (D). \[ \boxed{a=\frac{1}{\sqrt2},\ b=-\frac{1}{\sqrt2}} \]