The vector b = 3j + 4k is to be written as the sum of a vector b1 parallel to a = i +j and a vector b2 perpendicular to a. Then b1 is equal to

Question

The vector b = 3j + 4k is to be written as the sum of a vector b1 parallel to a = i +j and a vector b2 perpendicular to a. Then b1 is equal to (A) (3/2)( vec i+ vec j) (B) (2/3)( veci+ vecj) (C) (1/2)( vec i+ vecj) (D) (1/3)( vec i+ vec j)

Tardigrade Admin · Accepted Answer

Since, b 1 | a, therefore b 1-a( i + j ) b 2= b - b 1=(3-a) i -a j +4 k Also, b 2 ⋅ a =0 ⇒ (3-a)-a ⇒ a=(3/2) Hence, b 1=(3/2)( i + j )

Q. The vector $b = 3 j + 4 k$ is to be written as the sum of a vector $b_{1}$ parallel to $a = i + j$ and a vector $b_{2}$ perpendicular to $a$ . Then $b_{1}$ is equal to

$\frac{3}{2} (i + j)$

$\frac{2}{3} (i + j)$

$\frac{1}{2} (i + j)$

$\frac{1}{3} (i + j)$

Since, $b_{1} ∥ a$ , therefore $b_{1} - a (i + j)$
$b_{2} = b - b_{1} = (3 - a) i - aj + 4 k$
Also, $b_{2} \cdot a = 0$
$\Rightarrow (3 - a) - a \Rightarrow a = \frac{3}{2}$
Hence, $b_{1} = \frac{3}{2} (i + j)$

Q. The vector b=3j+4k is to be written as the sum of a vector b1​ parallel to a=i+j and a vector b2​ perpendicular to a. Then b1​ is equal to

23​(i+j​)

32​(i+j​)

21​(i+j​)

31​(i+j​)