Question

Brave that the vectay 7 = 4↑ +37 +R
and B = 12 7+ ai +3k are parallel
each other a​

Answer 1

Given :-

◉ Vector A = 4i + 3j + k

◉ Vector B = 12i + 9j + 3k

To Prove :-

◉ Vector A is parallel to Vector B

Solution :-

Let us find the magnitude of each of the two vectors.

⇒ |A| = √[ (Ax)² + (Ay)² + (Az)² ]

Where,

• Ax = X component of A
• Ay = Y component of A
• Az = Z component of A

⇒ |A| = √(4² + 3² + 1²)

⇒ |A| = √(16 + 9 + 1)

⇒ |A| = √26 ...(1)

Similarly,

⇒ |B| = √[ (Bx)² + (By)² + (Bz)² ]

Where,

• Bx = X component of B
• By = Y component of B
• Bz = Z component of B

⇒ |B| = √(12² + 9² + 3²)

⇒ |B| = √(144 + 81 + 9)

⇒ |B| = √234

⇒ |B| = 3√26 ...(2)

Adding (1) & (2),

⇒ |A| + |B| = √26 + 4√26 [ from (1) & (2) ]

⇒ |A| + |B| = 4√26

Now, Let us find the resultant of A & B,

⇒ A + B = (4 + 12)i + (3 + 9)j + (1+3)k

⇒ R = 16i + 12j + 4k [ A + B = R ]

So,

⇒ |R| = √[ (Rx)² + (Ry)² + (Rz)² ]

Where,

• Rx = X component of R
• Ry = Y component of R
• Rz = Z component of R

⇒ |R| = √(16² + 12² + 4²)

⇒ |R| = √(256 + 144 + 16)

⇒ |R| = √416

⇒ |R| = 4√26

We observe that |R| = |A| + |B|

It is only possible when the vectors are parallel to each other. In this case, The Magnitude of the two vectors gets added directly.

Hence, Proved.