Question

# In a right-angled triangle ABC, if the hypotenuse AB = p, then what is $$\overrightarrow{AB}. \overrightarrow{AC} + \overrightarrow{BC}. \overrightarrow{BA} + \overrightarrow{CA} . \overrightarrow{CB}$$ equal to?

A
p
B
p2
C
2p2
D
p22

Solution

## The correct option is B $$p^2$$$$\overrightarrow{AB}.\overrightarrow{AC}+\overrightarrow{BC}.\overrightarrow{BA}+\overrightarrow{CA}.\overrightarrow{CB}$$$$=\overrightarrow{AB}.\overrightarrow{AC}-\overrightarrow{BC}.\overrightarrow{AB}+0$$            ...($$\overrightarrow{CA}{\bot }\overrightarrow{CB}$$)$$=\overrightarrow{AB}.\overrightarrow{AC}+\overrightarrow{CB}.\overrightarrow{AB}$$$$=\overrightarrow{AB}.(\overrightarrow{AC}+\overrightarrow{CB})$$$$=\overrightarrow{AB}.(\overrightarrow{AB})$$$$=|\overrightarrow{AB}|^2$$$$=p^2$$Answer is option (B)Mathematics

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