Question:
The component of a vector
\[
\vec{P}=3\hat{i}+4\hat{j}
\]
along the direction
\[
\hat{i}+2\hat{j}
\]
is
The component of a vector
\[
\vec{P}=3\hat{i}+4\hat{j}
\]
along the direction
\[
\hat{i}+2\hat{j}
\]
is
Show Hint
The scalar component of vector \(\vec{A}\) along vector \(\vec{B}\) is
\[
\vec{A}\cdot \hat{B}
=
\frac{\vec{A}\cdot \vec{B}}{|\vec{B}|}.
\]
Updated On: Jun 22, 2026
- \(\frac{8}{\sqrt{5}}\)
- \(\frac{11}{\sqrt{5}}\)
- \(\frac{11}{2}\)
- \(\sqrt{10}\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Write the given vector and direction vector.
Given vector is \[ \vec{P}=3\hat{i}+4\hat{j} \] The direction vector is \[ \vec{a}=\hat{i}+2\hat{j} \]
Step 2: Find the unit vector along the given direction.
Magnitude of \(\vec{a}\) is \[ |\vec{a}|=\sqrt{1^2+2^2} \] \[ |\vec{a}|=\sqrt{5} \] Therefore, unit vector along \(\vec{a}\) is \[ \hat{a}=\frac{\hat{i}+2\hat{j}}{\sqrt{5}} \]
Step 3: Find the scalar component.
The component of \(\vec{P}\) along \(\vec{a}\) is \[ \vec{P}\cdot \hat{a} \] \[ =(3\hat{i}+4\hat{j})\cdot \frac{\hat{i}+2\hat{j}}{\sqrt{5}} \] \[ =\frac{3(1)+4(2)}{\sqrt{5}} \] \[ =\frac{3+8}{\sqrt{5}} \] \[ =\frac{11}{\sqrt{5}} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{11}{\sqrt{5}}} \]
Given vector is \[ \vec{P}=3\hat{i}+4\hat{j} \] The direction vector is \[ \vec{a}=\hat{i}+2\hat{j} \]
Step 2: Find the unit vector along the given direction.
Magnitude of \(\vec{a}\) is \[ |\vec{a}|=\sqrt{1^2+2^2} \] \[ |\vec{a}|=\sqrt{5} \] Therefore, unit vector along \(\vec{a}\) is \[ \hat{a}=\frac{\hat{i}+2\hat{j}}{\sqrt{5}} \]
Step 3: Find the scalar component.
The component of \(\vec{P}\) along \(\vec{a}\) is \[ \vec{P}\cdot \hat{a} \] \[ =(3\hat{i}+4\hat{j})\cdot \frac{\hat{i}+2\hat{j}}{\sqrt{5}} \] \[ =\frac{3(1)+4(2)}{\sqrt{5}} \] \[ =\frac{3+8}{\sqrt{5}} \] \[ =\frac{11}{\sqrt{5}} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{11}{\sqrt{5}}} \]
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