Question

If \(\overset{⃗}{a}+\overset{⃗}{b}+\overset{⃗}{c} = \overset{⃗}{0}\), \(|\overset{⃗}{a}| = |\overset{⃗}{b}| = |\overset{⃗}{c}| = 3\) and \(θ\) is the angle between \(\overset{⃗}{b}\) and \(\overset{⃗}{c}\) then \(tan^2θ+cot^2θ =\)

• A. \(\frac{2}{3}\)
• B. \(\frac{5}{3}\)
• C. \(\frac{8}{3}\)
• D. \(\frac{10}{3}\)

Hint:

Square the relation a = -(b + c) to find the angle between b and c.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Since \(\vec a + \vec b + \vec c = \vec 0\), we have \(\vec a = -(\vec b + \vec c)\). Square both sides to get a relation with the dot product.

Step 2: Find cos theta
\[ |\vec a|^2 = |\vec b|^2 + |\vec c|^2 + 2\vec b\cdot\vec c \Rightarrow 9 = 9 + 9 + 2\cdot 9\cos\theta \]
\[ \cos\theta = -\frac12 \Rightarrow \theta = 120^{\circ} \]

Step 3: Evaluate
\(\tan 120^{\circ} = -\sqrt3\), so \(\tan^2\theta = 3\), and \(\cot^2\theta = \frac13\).
\[ \tan^2\theta + \cot^2\theta = 3 + \frac13 = \frac{10}{3} \]
Option (C) 8/3 would be 3 minus 1/3, and (B) comes from a wrong angle.

Final Answer:
The value is \(\frac{10}{3}\), option (D). \[ \boxed{\frac{10}{3}} \]