First slide

Vectors

Question

Consider three vectors $A = \hat{i} + \hat{j} - 2 \hat{k}, B = \hat{i} - \hat{j} + \hat{k}$ and $C = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ . A vector X of the form $α$ A + $β$ B ( $α$ and $β$ are numbers) is perpendicular to C. The ratio of $α$ and $β$ is

Moderate

A

1 : 1
B

2 : 1
C

$-$ 1 : 1
D

3 : 1

Solution

Vector X of the form $α$ A + $β$ B.
$X = α A + β B$
$= α (\hat{i} + \hat{j} - 2 \hat{k}) + β (\hat{i} - \hat{j} + \hat{k})$
$X = \hat{i} (α + β) + \hat{j} (α - β) + \hat{k} (- 2 α + β)$
A vector X is perpendicular to C, i.e. X · C = 0
$[\hat{i} (α + β) + \hat{j} (α - β) + \hat{k} (- 2 α + β)] \cdot [2 \hat{i} - 3 \hat{j} + 4 \hat{k}] = 0$
$\Rightarrow 2 (α + β) - 3 (α - β) + 4 (- 2 α + β) = 0$
$\Rightarrow 2 α + 2 β - 3 α + 3 β - 8 α + 4 β = 0$
$\Rightarrow - 9 α + 9 β = 0$
or $α = β \Rightarrow \frac{α}{β} = \frac{1}{1}$
or $α : β = 1 : 1$

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