For three unit vectors a , b , c satisfying | a - b | 2 + | b - c | 2 + | c - a | 2 = 9 and | 2 + k b + k c | = 3 , the positive value of k is [2026]

Question

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2} = 9 and | 2 \vec{a} + k \vec{b} + k \vec{c} | = 3,$ the positive value of k is [2026]

Smart Achievers · Accepted Answer

(4)

$| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2} = 9$

$\Rightarrow \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = - \frac{3}{2}$

$\Rightarrow \vec{a} + \vec{b} + \vec{c} = 0 \Rightarrow \vec{b} + \vec{c} = - \vec{a}$

$| 2 \vec{a} + k (\vec{b} + \vec{c}) |$ $= 3$

$| \vec{a} (2 - k) |$ $= 3$

$k = 5 or - 1$

$Positive value of k is 5$

Solution

Q.
For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2} = 9 and | 2 \vec{a} + k \vec{b} + k \vec{c} | = 3,$ the positive value of k is [2026]

1 4

2 6

3 3

4 5

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Solution

Q. For three unit vectors a→,b→,c→ satisfying |a→-b→|2+|b→-c→|2+|c→-a→|2=9 and |2a→+kb→+kc→|=3, the positive value of k is [2026]

1 4

2 6

3 3

4 5

Ans. (4) |a→-b→|2+|b→-c→|2+|c→-a→|2=9 ⇒a→·b→+b→·c→+c→·a→=-32 ⇒a→+b→+c→=0⇒b→+c→=-a→ |2a→+k(b→+c→)|=3 |a→(2-k)|=3 k=5 or -1 Positive value of k is 5

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Q.
For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2} = 9 and | 2 \vec{a} + k \vec{b} + k \vec{c} | = 3,$ the positive value of k is [2026]