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For what value of m are the vector 2i – 3j + 4k, i + 2j – k and mi – j + 2k coplanar?
- a)0
- b)5/3
- c)1
- d)8/5
Correct answer is option 'D'. Can you explain this answer?
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For what value of m are the vector 2i 3j + 4k, i + 2j k and mi j + ...
Solution:
We know that three vectors are coplanar if their scalar triple product is zero.
Let the given vectors be A = 2i + 3j + 4k, B = i + 2j + k, and C = mi + j + 2k.
The scalar triple product of these vectors is given as:
A.(B x C) = 0
where B x C is the cross product of vectors B and C.
B x C = (2i - m j + k) x (i + j + 2k)
= 2i x i + 2i x j + 2i x 2k - m j x i - m j x j - m j x 2k + k x i + k x j + k x 2k
= 0i - 6mj + (2 - m)k
Therefore,
A.(B x C) = A.(-6mj + (2 - m)k)
= -18mj + (8 - 2m)k
= 0 (as A.(B x C) = 0 for coplanar vectors)
Comparing the coefficients of j and k, we get:
-18m = 0
8 - 2m = 0
Solving these equations, we get:
m = 8/5
Therefore, the value of m for which the given vectors are coplanar is 8/5.
Hence, the correct answer is option D.
We know that three vectors are coplanar if their scalar triple product is zero.
Let the given vectors be A = 2i + 3j + 4k, B = i + 2j + k, and C = mi + j + 2k.
The scalar triple product of these vectors is given as:
A.(B x C) = 0
where B x C is the cross product of vectors B and C.
B x C = (2i - m j + k) x (i + j + 2k)
= 2i x i + 2i x j + 2i x 2k - m j x i - m j x j - m j x 2k + k x i + k x j + k x 2k
= 0i - 6mj + (2 - m)k
Therefore,
A.(B x C) = A.(-6mj + (2 - m)k)
= -18mj + (8 - 2m)k
= 0 (as A.(B x C) = 0 for coplanar vectors)
Comparing the coefficients of j and k, we get:
-18m = 0
8 - 2m = 0
Solving these equations, we get:
m = 8/5
Therefore, the value of m for which the given vectors are coplanar is 8/5.
Hence, the correct answer is option D.
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