Question

If are two collinear vectors, then which of the following are incorrect : A. , for some scalar λ B. C. the respective components of are proportional D. both the vectors have same direction, but different magnitudes

Answer 1

The given vectors are collinear.

For option (A),

Let’s assume two collinear vectors,

a → =1 i ^ +1 j ^ +1 k ^

b → =2 i ^ +2 j ^ +2 k ^ =2( i ^ + j ^ + k ^ ) =2 a →

So, b → =λ a →

Here,

λ=2

Thus, option (A) is correct.

For option (B),

Let’s assume,

a → =1 i ^ +1 j ^ +1 k ^

From the given condition,

b → =1 i ^ +1 j ^ +1 k ^

Or,

b → =− i ^ − j ^ − k ^

For both cases of b → , a → and b → are collinear.

Thus, option (B) is correct.

For option (C),

Let’s assume two collinear vectors, a → =1 i ^ +1 j ^ +1 k ^ and b → =2 i ^ +2 j ^ +2 k ^ .

Ratios of their respective components are ( 1 2 , 1 2 , 1 2 ).

Thus, the respective components of a → and b → are proportional.

Thus, option (C) is correct.

For option (D),

When two or more vectors are parallel to same line, then these vectors are collinear, they can have different magnitudes and opposite directions .

Vectors a → and b → have different directions.

Thus, option (D) is incorrect.