Question 2 Exercise 3.4

Solutions of Question 2 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Show in two different ways that the vectors aa and bb are parallel to a=ˆi+2ˆj3ˆk,b=2ˆi4ˆj+a=^i+2^j3^k,b=2^i4^j+ 6ˆk6^k

First Way a×b=|ˆiˆjˆk123246|=(1212)ˆi(6+6)ˆj+(44)ˆka×b=0.ab.

Second Way ab=(ˆi+2ˆj3ˆk)(2ˆi4ˆj+6ˆk)ab=1(2)+2(4)3(6)ab=28. Also |a|=(1)2+(2)2+(3)2|a|=14|b|=(2)2+(4)2+(6)2|b|=56cosθ=ab|a|=281456θ=cos1(2821414)θ=cos1(1)=180.ab.

Show in two different ways that the vectors a and b are parallel to a=3ˆi+6ˆj9ˆk, b=ˆi+2ˆj3ˆk

First Way a×b=|ˆiˆjˆk369123|=(18+18)ˆi+(9+9)ˆj+(66)ˆka×b=0.ab.

Second Way ab=(3ˆi+6ˆj9ˆk)(ˆi+2ˆj3ˆk)ab=3(1)+6(2)9(3)ab=42 Also |a|=(3)2+(6)2+(9)2|a|=12¯6 and|b|=(1)2+(2)2+(3)2|b|=14. Now we know that
cosθ=abi|b|=4214126θ=cos1(42×4214×126)θ=cos1(17641764)θ=cos1(1)=0ab.