First slide

Dot product or scalar product of vectors

Question

If $A (1, 2, 3), B (2, 3, 4), C (3, 4, 5), D (4, 5, 6)$ are four points, then the projection of $\bar{B D} on \bar{A C}$ is

Moderate

A

$\sqrt{3}$
B

$2 \sqrt{3}$
C

$\sqrt{6}$
D

6

Solution

The vectors $\bar{B D} = 2 i + 2 j + 2 k$ and $\bar{A C} = 2 i + 2 j + 2 k$
The projection of $\bar{B D}$ on $\bar{A C}$ is $\frac{\bar{B D} \cdot \bar{A C}}{|\bar{A C}|}$
It implies that $\frac{\bar{B D} \cdot \bar{A C}}{|\bar{A C}|} = \frac{4 + 4 + 4}{\sqrt{4 + 4 + 4}} = \sqrt{12} = 2 \sqrt{3}$

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