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

• A. $\dfrac{63}{65}$
• B. $63$
• C. $\dfrac{63}{4225}$
• D. $\dfrac{63}{845}$

Hint:

The dot product of two unit vectors is the cosine of the angle between them. For vectors parallel to given vectors, first find the unit vectors by dividing by their magnitudes, then compute the dot product.

Answer 1

The Correct Option is A

To find the dot product of unit vectors $\hat{n}_1$ and $\hat{n}_2$ that are parallel to $5\hat{i} + 12\hat{j}$ and $3\hat{i} + 4\hat{j}$, we first determine the unit vectors and then compute their dot product.
- Vector 1: $5\hat{i} + 12\hat{j}$. Compute its magnitude:
\[ \begin{align} |5\hat{i} + 12\hat{j}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] The unit vector $\hat{n}_1$ parallel to $5\hat{i} + 12\hat{j}$ is: \[ \begin{align} \hat{n}_1 = \frac{5\hat{i} + 12\hat{j}}{13} = \frac{5}{13}\hat{i} + \frac{12}{13}\hat{j} \] - Vector 2: $3\hat{i} + 4\hat{j}$. Compute its magnitude: \[ \begin{align} |3\hat{i} + 4\hat{j}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] The unit vector $\hat{n}_2$ parallel to $3\hat{i} + 4\hat{j}$ is: \[ \begin{align} \hat{n}_2 = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \] - Dot Product: The dot product of two unit vectors $\hat{n}_1 \cdot \hat{n}_2$ is: \[ \begin{align} \hat{n}_1 \cdot \hat{n}_2 = \left(\frac{5}{13}\hat{i} + \frac{12}{13}\hat{j}\right) \cdot \left(\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}\right) = \left(\frac{5}{13} \cdot \frac{3}{5}\right) + \left(\frac{12}{13} \cdot \frac{4}{5}\right) \] \[ \begin{align} = \frac{5 \times 3}{13 \times 5} + \frac{12 \times 4}{13 \times 5} = \frac{15}{65} + \frac{48}{65} = \frac{15 + 48}{65} = \frac{63}{65} \] The dot product matches option (1).
Thus, the correct answer is (1).