First slide

Dot product or scalar product of vectors

Question

If in a right-angled triangle ABC, the hypotenuse A B= p, the $\vec{AB} \cdot \vec{AC} + \vec{BC} \cdot \vec{BA} + \vec{CA} \cdot \vec{CB} =$

Moderate

A

2P²
B

$\frac{P^{2}}{2}$
C

P²
D

none of these

Solution

We have
$\begin{array}{l} \vec{AB} \cdot \vec{AC} + \vec{BC} \cdot \vec{BA} + \vec{CA} \cdot \vec{CB} \\ = (AB) (AC) \cos θ + (BC) (BA) \sin θ + 0 \\ = AB (ACcos θ + BCsin θ) \\ = AB (\frac{(AC)^{2}}{AB} + \frac{(BC)^{2}}{AB}) \\ = {AC}^{2} + {BC}^{2} = {AB}^{2} = p^{2} \end{array}$

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