The value of P so that the vectors 2 hat i - hat j + hat k, hat i + 2 hat j - 3 hat k and 3 hat i + P hat j + 5 hat k are coplanar should be

Question

The value of P so that the vectors 2 hat i - hat j + hat k, hat i + 2 hat j - 3 hat k and 3 hat i + P hat j + 5 hat k are coplanar should be (A) 16 (B) -4 (C) 4 (D) -8

Tardigrade Admin · Accepted Answer

For coplanarity 35mm | 2 -1 1 1 2 -3 3 P 5 | = 0 or 2(10 + 3P) + 1(5 + 9) +1(P - 6)=0 or 20 + 6P + 5 + 9 + P - 6 = 0 or 7P + 34 - 6 = 0 or 7P + 28 = 0 or 7P = -28 ⇒ P = - (28/7) = -4

Q. The value of P so that the vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $3 \hat{i} + P \hat{j} + 5 \hat{k}$ are coplanar should be

16

-4

4

-8

For coplanarity
$35 mm ∣ ∣ 213 - 1 2 P 1 - 3 5 ∣ ∣ = 0$
or $2 (10 + 3 P) + 1 (5 + 9) + 1 (P - 6) = 0$
or $20 + 6 P + 5 + 9 + P - 6 = 0$
or $7 P + 34 - 6 = 0$
or $7 P + 28 = 0$
or $7 P = - 28$
$\Rightarrow P = - \frac{28}{7} = - 4$

Q. The value of P so that the vectors 2i^−j^​+k^,i^+2j^​−3k^ and 3i^+Pj^​+5k^ are coplanar should be

16

-4

4

-8

For coplanarity 35mm∣∣​213​−12P​1−35​∣∣​=0 or 2(10+3P)+1(5+9)+1(P−6)=0 or 20+6P+5+9+P−6=0 or 7P+34−6=0 or 7P+28=0 or 7P=−28 ⇒P=−728​=−4

Q. The value of P so that the vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $3 \hat{i} + P \hat{j} + 5 \hat{k}$ are coplanar should be

For coplanarity
$35 mm ∣ ∣ 213 - 1 2 P 1 - 3 5 ∣ ∣ = 0$
or $2 (10 + 3 P) + 1 (5 + 9) + 1 (P - 6) = 0$
or $20 + 6 P + 5 + 9 + P - 6 = 0$
or $7 P + 34 - 6 = 0$
or $7 P + 28 = 0$
or $7 P = - 28$
$\Rightarrow P = - \frac{28}{7} = - 4$