First slide

Dot product or scalar product of vectors

Question

$\vec{a} and \vec{c}$ are unit vector and $| \vec{b} | = 4$ .the angle between $\vec{a} and \vec{c} is \cos^{- 1} (1 / 4) and \vec{b} - 2 \vec{c} = λ \vec{a}$ the value of $λ$ is

Moderate

A

3,- 4
B

$\frac{1}{4}, \frac{3}{4}$
C

-3, 4
D

$\frac{- 1}{4}, \frac{3}{4}$

Solution

$\begin{array}{l} \vec{b} - 2 \vec{c} = λ \vec{a} \\ or \vec{b} = 2 \vec{c} + λ \vec{a} \\ or | \vec{b} |^{2} = | 2 \vec{c} + λ \vec{a} |^{2} \end{array}$
$\begin{array}{l} \Rightarrow 16 = 4 | \vec{c} |^{2} + λ^{2} | \vec{a} |^{2} + 4 λ \vec{a} \cdot \vec{c} \\ = 4 + λ^{2} + 4 λ \frac{1}{4} \\ or λ^{2} + λ - 12 = 0 \\ or λ = 3, - 4 \end{array}$

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