Question:
The dot product of unit vectors \(\hat{n}_1\) and \(\hat{n}_2\) that are parallel to
\[
5\hat{i}+12\hat{j}
\]
and
\[
3\hat{i}+4\hat{j}
\]
respectively is:
The dot product of unit vectors \(\hat{n}_1\) and \(\hat{n}_2\) that are parallel to
\[
5\hat{i}+12\hat{j}
\]
and
\[
3\hat{i}+4\hat{j}
\]
respectively is:
Show Hint
To find the unit vector parallel to a vector, divide the vector by its magnitude. Then use the dot product formula component-wise.
Updated On: Jun 24, 2026
- \(\dfrac{63}{65}\)
- \(63\)
- \(\dfrac{63}{4225}\)
- \(\dfrac{63}{845}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Find the unit vector parallel to \(5\hat{i}+12\hat{j}\).
Let \[ \vec{a}=5\hat{i}+12\hat{j} \] Magnitude of \(\vec{a}\) is \[ |\vec{a}|=\sqrt{5^2+12^2} \] \[ =\sqrt{25+144} \] \[ =\sqrt{169} \] \[ =13 \] Therefore, \[ \hat{n}_1=\frac{5\hat{i}+12\hat{j}}{13} \] \[ \hat{n}_1=\frac{5}{13}\hat{i}+\frac{12}{13}\hat{j} \]
Step 2: Find the unit vector parallel to \(3\hat{i}+4\hat{j}\).
Let \[ \vec{b}=3\hat{i}+4\hat{j} \] Magnitude of \(\vec{b}\) is \[ |\vec{b}|=\sqrt{3^2+4^2} \] \[ =\sqrt{9+16} \] \[ =\sqrt{25} \] \[ =5 \] Therefore, \[ \hat{n}_2=\frac{3\hat{i}+4\hat{j}}{5} \] \[ \hat{n}_2=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j} \]
Step 3: Find the dot product.
\[ \hat{n}_1\cdot \hat{n}_2 = \left(\frac{5}{13}\hat{i}+\frac{12}{13}\hat{j}\right) \cdot \left(\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}\right) \] \[ = \frac{5}{13}\cdot \frac{3}{5} + \frac{12}{13}\cdot \frac{4}{5} \] \[ = \frac{15}{65}+\frac{48}{65} \] \[ = \frac{63}{65} \]
Step 4: Final conclusion.
Hence, \[ \boxed{\frac{63}{65}} \]
Let \[ \vec{a}=5\hat{i}+12\hat{j} \] Magnitude of \(\vec{a}\) is \[ |\vec{a}|=\sqrt{5^2+12^2} \] \[ =\sqrt{25+144} \] \[ =\sqrt{169} \] \[ =13 \] Therefore, \[ \hat{n}_1=\frac{5\hat{i}+12\hat{j}}{13} \] \[ \hat{n}_1=\frac{5}{13}\hat{i}+\frac{12}{13}\hat{j} \]
Step 2: Find the unit vector parallel to \(3\hat{i}+4\hat{j}\).
Let \[ \vec{b}=3\hat{i}+4\hat{j} \] Magnitude of \(\vec{b}\) is \[ |\vec{b}|=\sqrt{3^2+4^2} \] \[ =\sqrt{9+16} \] \[ =\sqrt{25} \] \[ =5 \] Therefore, \[ \hat{n}_2=\frac{3\hat{i}+4\hat{j}}{5} \] \[ \hat{n}_2=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j} \]
Step 3: Find the dot product.
\[ \hat{n}_1\cdot \hat{n}_2 = \left(\frac{5}{13}\hat{i}+\frac{12}{13}\hat{j}\right) \cdot \left(\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}\right) \] \[ = \frac{5}{13}\cdot \frac{3}{5} + \frac{12}{13}\cdot \frac{4}{5} \] \[ = \frac{15}{65}+\frac{48}{65} \] \[ = \frac{63}{65} \]
Step 4: Final conclusion.
Hence, \[ \boxed{\frac{63}{65}} \]
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