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:

To find the unit vector parallel to a vector, divide the vector by its magnitude. Then use the dot product formula component-wise.

Answer 1

The Correct Option is A

Step 1: Understand what the problem asks.
We need unit vectors $\hat{n}_1$ and $\hat{n}_2$ parallel to the given vectors, then find their dot product.
The dot product of two unit vectors equals $\cos\theta$ where $\theta$ is the angle between the original vectors.

Step 2: State the dot product formula using magnitudes.
For any two vectors $\vec{a}$ and $\vec{b}$:
\[ \hat{n}_1 \cdot \hat{n}_2 = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \]

Step 3: Compute the dot product of the two original vectors.
Let $\vec{a} = 5\hat{i} + 12\hat{j}$ and $\vec{b} = 3\hat{i} + 4\hat{j}$.
\[ \vec{a} \cdot \vec{b} = (5)(3) + (12)(4) = 15 + 48 = 63 \]

Step 4: Find the magnitude of each vector.
\[ |\vec{a}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] \[ |\vec{b}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 5: Compute the dot product of the unit vectors.
\[ \hat{n}_1 \cdot \hat{n}_2 = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{63}{13 \times 5} = \frac{63}{65} \]

Step 6: State the final answer.
\[ \boxed{\dfrac{63}{65}} \]