If the two adjacent sides of two rectangles are represented by the vectors →p=5→a−3→b;→q=−→a−2→b and →r=−4→a−→b;→s=−→a+→b respectively, then the angle between the vectors →x=13(→p+→r+→s) and →y=15(→r+→s).
If the two adjacent sides of two rectangles are represented by the vectors →p=5→a−3→b;→q=−→a−2→b and →r=−4→a−→b;→s=−→a+→b respectively, then the angle between the vectors →x=13(→p+→r+→s) and →y=15(→r+→s).
The correct option is B cos−1(195√43)
→p=5→a−3→b,→q=−→a−→2b,→r=−4→a−→b
and →s=−→a+→b
So, →p.→q=0 &
→r.→s=0(5→a−3→b).(−→a−→2b)=0−5∣∣→a∣∣2+6∣∣∣→b∣∣∣2−7∣∣→a∣∣.∣∣∣→b∣∣∣=0⟶(i)
& (−4→a−→b)(−→a+→b)=04∣∣→a∣∣2−3∣∣→a∣∣∣∣∣→b∣∣∣−∣∣∣→b∣∣∣2=0⟶(ii)→x=13(→p+→q+→s)=13(5→a−3→b−4→a−→b−→a+→b)=−3→b3=−→b
→y=15(−4→a−→b−→a+→b)=−→a
Angle between them
cosθ=→a→b|a||b|
From (i)&(ii) putting value of (ii)∣∣∣→b∣∣∣2in(i)
−5∣∣→a∣∣2+24∣∣→a∣∣2−8∣∣→a∣∣∣∣∣→b∣∣∣−7∣∣→a∣∣∣∣∣→b∣∣∣=0∣∣→a∣∣2=2519∣∣→a∣∣∣∣∣→b∣∣∣∣∣→a∣∣=√2519√∣∣→a∣∣∣∣∣→b∣∣∣∣∣∣→b∣∣∣=√4×2519−3√∣∣→a∣∣∣∣∣→b∣∣∣=√100−5719√∣∣→a∣∣∣∣∣→b∣∣∣=√4319√∣∣→a∣∣∣∣∣→b∣∣∣cosθ=∣∣→a∣∣.∣∣∣→b∣∣∣√2519√4319∣∣→a∣∣∣∣∣→b∣∣∣cosθ=195√43θ=cos−1(195√43)
→p=5→a−3→b,→q=−→a−→2b,→r=−4→a−→b
and →s=−→a+→b
So, →p.→q=0 &
→r.→s=0(5→a−3→b).(−→a−→2b)=0−5∣∣→a∣∣2+6∣∣∣→b∣∣∣2−7∣∣→a∣∣.∣∣∣→b∣∣∣=0⟶(i)
& (−4→a−→b)(−→a+→b)=04∣∣→a∣∣2−3∣∣→a∣∣∣∣∣→b∣∣∣−∣∣∣→b∣∣∣2=0⟶(ii)→x=13(→p+→q+→s)=13(5→a−3→b−4→a−→b−→a+→b)=−3→b3=−→b
→y=15(−4→a−→b−→a+→b)=−→a
Angle between them
cosθ=→a→b|a||b|
From (i)&(ii) putting value of (ii)∣∣∣→b∣∣∣2in(i)
−5∣∣→a∣∣2+24∣∣→a∣∣2−8∣∣→a∣∣∣∣∣→b∣∣∣−7∣∣→a∣∣∣∣∣→b∣∣∣=0∣∣→a∣∣2=2519∣∣→a∣∣∣∣∣→b∣∣∣∣∣→a∣∣=√2519√∣∣→a∣∣∣∣∣→b∣∣∣∣∣∣→b∣∣∣=√4×2519−3√∣∣→a∣∣∣∣∣→b∣∣∣=√100−5719√∣∣→a∣∣∣∣∣→b∣∣∣=√4319√∣∣→a∣∣∣∣∣→b∣∣∣cosθ=∣∣→a∣∣.∣∣∣→b∣∣∣√2519√4319∣∣→a∣∣∣∣∣→b∣∣∣cosθ=195√43θ=cos−1(195√43)
