Question

If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}+\hat{k}$ and $\theta$ is the angle between them, then $\tan\theta=$

• A. $\frac{\sqrt{38}}{4}$
• B. $\frac{\sqrt{26}}{4}$
• C. $\frac{\sqrt{26}}{5}$
• D. $\frac{\sqrt{26}}{6}$
• E. $\frac{\sqrt{38}}{6}$

Hint:

Trigonometry Tip: Drawing a quick right triangle to find missing ratios is almost always faster and less prone to algebraic errors than wrestling with complex trigonometric identities like $\tan \theta = \sqrt{\sec^2\theta - 1}$.

Answer 1

The Correct Option is B

Concept:
The angle $\theta$ between two vectors can be found primarily using the dot product formula: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$. Once $\cos \theta$ is identified, we can construct a right triangle to find the corresponding $\tan \theta$ value.

Step 1: Calculate the dot product of the vectors.
Multiply the corresponding components of $\vec{a}$ and $\vec{b}$, then sum them: $$\vec{a} \cdot \vec{b} = (1)(2) + (1)(3) + (-1)(1)$$ $$\vec{a} \cdot \vec{b} = 2 + 3 - 1 = 4$$

Step 2: Calculate the magnitudes of both vectors.
Find the lengths using the standard vector magnitude formula: $$|\vec{a}| = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3}$$ $$|\vec{b}| = \sqrt{2^2 + 3^2 + 1^2} = \sqrt{4 + 9 + 1} = \sqrt{14}$$

Step 3: Find the cosine of the angle.
Substitute the dot product and magnitudes into the geometric angle formula: $$\cos \theta = \frac{4}{\sqrt{3}\sqrt{14}}$$ $$\cos \theta = \frac{4}{\sqrt{42}}$$

Step 4: Determine the opposite side using a right triangle.
Using a right triangle where the adjacent side is $4$ and the hypotenuse is $\sqrt{42}$, we find the opposite side using the Pythagorean theorem ($a^2 + b^2 = c^2$): $$\text{Opposite} = \sqrt{(\sqrt{42})^2 - 4^2} = \sqrt{42 - 16} = \sqrt{26}$$

Step 5: Calculate the tangent of the angle.
Since the tangent of an angle is the ratio of the opposite side to the adjacent side ($\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$): $$\tan \theta = \frac{\sqrt{26}}{4}$$ Hence the correct answer is (B) $\frac{\sqrt{26{4}$}.