Let L1: vec r = hat i - hat j -10 hat k +λ(2 hat i -3 hat j +8 hat k ) and L 2: vec r =4 hat i -3 hat j - hat k +μ( hat i -4 hat j +7 hat k ) represent two lines in R 3, then which one of the following is incorrect?

Question

Let L1: vec r = hat i - hat j -10 hat k +λ(2 hat i -3 hat j +8 hat k ) and L 2: vec r =4 hat i -3 hat j - hat k +μ( hat i -4 hat j +7 hat k ) represent two lines in R 3, then which one of the following is incorrect? (A) L1 is parallel to the vector 4 hati-6 hatj+16 hatk. (B) L2 is parallel to the vector - hati+4 hatj-7 hatk. (C) L1 and L2 are coplanar. (D) Angle between the lines L 1 and L 2 is cos -1((70/11 √7)).

Tardigrade Admin · Accepted Answer

L 1 and L 2 are intersecting lines. The position vector of their point of intersection is 5 hat i -7 hat j +6 hat k (For λ=2 or μ=1 ). Also, angle between L 1 and L 2=(70/11 √42).

Q. Let $L_{1} : r = \hat{i} - \hat{j} - 10 \hat{k} + λ (2 \hat{i} - 3 \hat{j} + 8 \hat{k})$ and $L_{2}$ : $r = 4 \hat{i} - 3 \hat{j} - \hat{k} + μ (\hat{i} - 4 \hat{j} + 7 \hat{k})$ represent two lines in $R^{3}$ , then which one of the following is incorrect?

$L_{1}$ is parallel to the vector $4 \hat{i} - 6 \hat{j} + 16 \hat{k}$ .

$L_{2}$ is parallel to the vector $- \hat{i} + 4 \hat{j} - 7 \hat{k}$ .

$L_{1}$ and $L_{2}$ are coplanar.

Angle between the lines $L_{1}$ and $L_{2}$ is $cos^{- 1} (\frac{70}{11 7})$ .

$L_{1}$ and $L_{2}$ are intersecting lines.
The position vector of their point of intersection is
$5 \hat{i} - 7 \hat{j} + 6 \hat{k}$ (For $λ = 2$ or $μ = 1$ ).
Also, angle between $L_{1}$ and $L_{2} = \frac{70}{11 42}$ .

Q. Let L1​:r=i^−j^​−10k^+λ(2i^−3j^​+8k^) and L2​ : r=4i^−3j^​−k^+μ(i^−4j^​+7k^) represent two lines in R3, then which one of the following is incorrect?

L1​ is parallel to the vector 4i^−6j^​+16k^.

L2​ is parallel to the vector −i^+4j^​−7k^.

L1​ and L2​ are coplanar.

Angle between the lines L1​ and L2​ is cos−1(117​70​).