Question

If \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are unit vectors and \(|\overline{\mathrm{a}}|=7\), \(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \times \overline{\mathrm{a}})=\frac{1}{2} \overline{\mathrm{a}}\), then angle between the vectors \(\bar{a}\) and \(\bar{c}\) and angle between the vectors \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are respectively

• A. \(90^{\circ},60^{\circ}\)
• B. \(30^{\circ},60^{\circ}\)
• C. \(90^{\circ},120^{\circ}\)
• D. \(45^{\circ},90^{\circ}\)

Hint:

Expand each triple product with the vector triple product formula and compare the coefficients of the vectors.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Use \(\bar a\times(\bar b\times\bar c)=\bar b(\bar a\cdot\bar c)-\bar c(\bar a\cdot\bar b)\).

Step 2: Expand both terms
\(\bar a\times(\bar b\times\bar c)=\bar b(\bar a\cdot\bar c)-\bar c(\bar a\cdot\bar b)\)
\(\bar b\times(\bar c\times\bar a)=\bar c(\bar b\cdot\bar a)-\bar a(\bar b\cdot\bar c)\)

Step 3: Add them
The \(\bar c\) terms cancel, so the sum is \(\bar b(\bar a\cdot\bar c)-\bar a(\bar b\cdot\bar c)=\dfrac12\bar a\).
Since \(\bar a\) and \(\bar b\) are independent, \(\bar a\cdot\bar c=0\) and \(-(\bar b\cdot\bar c)=\dfrac12\).

Step 4: Calculation
\(\bar a\cdot\bar c=0\) gives the angle between \(\bar a\) and \(\bar c\) as \(90^{\circ}\). \(\bar b\cdot\bar c=-\dfrac12\) with unit vectors gives \(\cos\theta=-\dfrac12\), so \(\theta=120^{\circ}\). This is option (C).

Final Answer:
The angles are 90 degrees and 120 degrees. \[ \boxed{\text{(C) }90^{\circ},\,120^{\circ}} \]