First slide

Algebra of vectors

Question

If $\vec{b}$ is a vector whose initial point divides the join of $5 \hat{i} and 5 \hat{j}$ in the ratio k:1 and whose terminal point is the origin and $| \vec{b} | \leq \sqrt{37}$ then k lies in the interval

Difficult

A

$[- 6, - 1 / 6]$
B

$(- \infty, - 6] \cup [- 1 / 6, \infty)$
C

[0, 6]
D

none of these

Solution

The point that divides $5 \hat{i} and 5 \hat{j}$ in the ratio of
$\begin{array}{l} k : 1 is \frac{(5 \hat{j}) k + (5 \hat{i}) 1}{k + 1} \\ ∴ \vec{b} = \frac{5 \hat{i} + 5 k \hat{j}}{k + 1} \end{array}$
$Also, | \vec{b} | \leq \sqrt{37}$
$\begin{aligned} \Rightarrow \frac{1}{k + 1} \sqrt{25 + 25 k^{2}} \leq \sqrt{37} \\ or 5 \sqrt{1 + k^{2}} \leq \sqrt{37} (k + 1) \end{aligned}$
Squaring both sides, we get
$\begin{array}{l} 25 (1 + k^{2}) \leq 37 (k^{2} + 2 k + 1) \\ or 6 k^{2} + 37 k + 6 \geq 0 or (6 k + 1) (k + 6) \geq 0 \\ k \in (- \infty, - 6] \cup [- \frac{1}{6}, \infty) \end{array}$

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