Let veca = 3 hati + 2 hatj + 2 hatk and vecb = hati + 2 hatj - 2 hatk be two vectors. If a vector perpendicular to both the vectors veca + vecb and veca - vecb has the magnitude 12 then one such vector is

Question

Let veca = 3 hati + 2 hatj + 2 hatk and vecb = hati + 2 hatj - 2 hatk be two vectors. If a vector perpendicular to both the vectors veca + vecb and veca - vecb has the magnitude 12 then one such vector is (A)  4 ( 2 hati + 2 hatj - hatk) (B)  4 ( - 2 hati - 2 hatj + hatk) (C)  4 ( 2 hati - 2 hatj - hatk) (D)  4 ( 2 hati + 2 hatj + hatk)

Tardigrade Admin · Accepted Answer

( veca + vecb) ×( veca - vecb) = 2 ( vecb × veca) = 2 | hati& hatj& hatk 1&2&-2 3&2&2| = 2 (8 hati -8 hatj +4 hatk) Required vector = ± 12 ((2 hati -2 hatj - hatk)/3) = ± 4 (2 hati - 2 hatj - hatk)

Q. Let $a = 3 \hat{i} + 2 \hat{j} + 2 \hat{k}$ and $b = \hat{i} + 2 \hat{j} - 2 \hat{k}$ be two vectors. If a vector perpendicular to both the vectors $a + b$ and $a - b$ has the magnitude $12$ then one such vector is

$4 (2 \hat{i} + 2 \hat{j} - \hat{k})$

$4 (- 2 \hat{i} - 2 \hat{j} + \hat{k})$

$4 (2 \hat{i} - 2 \hat{j} - \hat{k})$

$4 (2 \hat{i} + 2 \hat{j} + \hat{k})$

$(a + b) \times (a - b)$
$= 2 (b \times a)$
$= 2 ∣ ∣ \hat{i} 13 \hat{j} 22 \hat{k} - 2 2 ∣ ∣$
$= 2 (8 \hat{i} - 8 \hat{j} + 4 \hat{k})$
Required vector $= \pm 12 \frac{( 2 i ^ - 2 j ^ - k ^ )}{3}$
$= \pm 4 (2 \hat{i} - 2 \hat{j} - \hat{k})$

Q. Let a=3i^+2j^​+2k^ and b=i^+2j^​−2k^ be two vectors. If a vector perpendicular to both the vectors a+b and a−b has the magnitude 12 then one such vector is

4(2i^+2j^​−k^)

4(−2i^−2j^​+k^)

4(2i^−2j^​−k^)

4(2i^+2j^​+k^)