Vector Algebra — Questions & Answers

Sample questions from Vector Algebra

1. Q1. Assertion (A): [a b c] = 0 does not mean any vector is zero. Reason (R): [a b c] = 0 means the vectors are coplanar.

   • A.Both A and R are true, and R is the correct explanation of A
   • B.Both A and R are true, but R is not the correct explanation of A✓
   • C.A is true but R is false
   • D.A is false but R is true
   Solution

   B is correct. Both A and R are true. [a b c]=0 means coplanar (R), but coplanar non-zero vectors certainly exist (e.g., 3 vectors in the xy-plane). So A is true for the right reason, but R (coplanarity) explains the geometric meaning of A, not why any vector need not be zero — these are separate truths, R does not explain A causally.

2. Q2. If ABCD is a parallelogram with AB = a and AD = b, then AC (diagonal) is:

   • A.a − b
   • B.b − a
   • C.a + b✓
   • D.2a + b
   Solution

   C is correct. In parallelogram ABCD, AC = AB + BC = AB + AD = a + b (since BC = AD = b).

3. Q3. Assertion (A): For any three vectors a, b, c: [a+b, b+c, c+a] = 2[a b c]. Reason (R): The scalar triple product is linear in each argument.

   • A.Both A and R are true, and R is the correct explanation of A✓
   • B.Both A and R are true, but R is not the correct explanation of A
   • C.A is true but R is false
   • D.A is false but R is true
   Solution

   A is correct. Both A and R are true. Expanding [a+b, b+c, c+a] by linearity and using [a b c]=−[b a c] etc., only 2[a b c] survives. R (linearity) is the correct explanation.

4. Q4. Assertion (A): If a, b, c are non-coplanar, then [a + b, b + c, c + a] = 2[a b c]. Reason (R): The scalar triple product changes sign when any two of its vectors are interchanged.

   • A.Both A and R are true, and R is the correct explanation of A
   • B.Both A and R are true, but R is not the correct explanation of A✓
   • C.A is true but R is false
   • D.A is false but R is true
   Solution

   Both statements are true, but the expansion of [a + b, b + c, c + a] rests on linearity in each argument together with the vanishing of triple products having a repeated vector, not on antisymmetry alone.

5. Q5. If [a b c] = -4 for non-coplanar vectors, then [a x b, b x c, c x a] equals:

   • A.-16
   • B.-64
   • C.16✓
   • D.-4
   Solution

   The standard result is [a x b, b x c, c x a] = [a b c]^2, and (-4)^2 = 16.

Other Class 12 Mathematics chapters