If p=i-2 j+3 k and q=3 i+j+2 k, than a vector r which is linear combination of p and q and also perpendicular to q is

Question

If p=i-2 j+3 k and q=3 i+j+2 k, than a vector r which is linear combination of p and q and also perpendicular to q is (A) i+5 j-4 k (B) i-5 j+4 k (C) (-1/2)(i+5 j-4 k) (D) none of these

Tardigrade Admin · Accepted Answer

We have, r=p+λ q ⇒ r ⋅ q=p ⋅ q+λ q ⋅ q ∴ 0=7+14 λ (∵ p ⋅ q=3-2+6=7 and q ⋅ q=9+1+4=14) ⇒ λ=-(7/14)=-(1/2) ⋅ ∴ r=-(1/2)(i+5 j-4 k)

Q. If $p = i - 2 j + 3 k$ and $q = 3 i + j + 2 k$ , than a vector $r$ which is linear combination of $p$ and $q$ and also perpendicular to $q$ is

$i + 5 j - 4 k$

$i - 5 j + 4 k$

$\frac{- 1}{2} (i + 5 j - 4 k)$

none of these

We have, $r = p + λ q$
$\Rightarrow r \cdot q = p \cdot q + λ q \cdot q$
$∴ 0 = 7 + 14 λ$
$(∵ p \cdot q = 3 - 2 + 6 = 7$ and $q \cdot q = 9 + 1 + 4 = 14)$
$\Rightarrow λ = - \frac{7}{14} = - \frac{1}{2} \cdot$
$∴ r = - \frac{1}{2} (i + 5 j - 4 k)$

Q. If p=i−2j+3k and q=3i+j+2k, than a vector r which is linear combination of p and q and also perpendicular to q is

i+5j−4k

i−5j+4k

2−1​(i+5j−4k)