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Question

# Three vectors →P,→Q and →R are such that |→P|=|→Q|,|→R|=√2|→P|, and →P+→Q+→R=→0. The angles between →P and →Q,→Q and →R, and →P and →R are

A
90,135,135
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B
90,45,45
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C
45,90,90
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D
45,135,135
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Solution

## The correct option is A 90∘,135∘,135∘Vectors →P+→Q+→R=→0 Given: |→P|=|→Q| and |→R|=√2|→P| Let ϕ be the angle between vectors →P and →Q. So, using vector addition →P+→Q=−→R |→P+→Q|2=|−→R|2=|→R|2 |→P|2+|→Q|2+2|→P||→Q|×cosϕ=2|→P|2 Hence, cosϕ=0⇒ϕ=π2=90∘ As →P and →Q vectors of equal magnitude, −→R will be at π4 angle from both →P and →Q. So →R will be at 180∘ from - →R. Angle between →Q and →R =180∘−45∘=135∘ Angle between vectors →R and →P=135∘

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