Question

Which of the following statements are true? A. For any two vectors \(\vec{a}\) and \(\vec{b}\), \(|\vec{a}+\vec{b}|\leq |\vec{a}|+|\vec{b}|\).
B. Scalar product of two non-zero vectors might be zero.
C. For any two vectors \(\vec{a}\) and \(\vec{b}\), \(|\vec{a}\cdot \vec{b}|\leq |\vec{a}||\vec{b}|\).
D. Cross product of two vectors is not a vector.
E. Dot product of two vectors is a scalar.

• A. A, B, C only
• B. B, C, D, E only
• C. A, B, C, E only
• D. A, B, C, D only

Hint:

Dot product gives a scalar, while cross product gives a vector.

Answer 1

The Correct Option is C

Concept:
This question is based on basic properties of vectors, dot product, and cross product.

Step 1: Check statement A.
By triangle inequality: \[ |\vec{a}+\vec{b}|\leq |\vec{a}|+|\vec{b}| \] So, A is true.

Step 2: Check statement B.
If two non-zero vectors are perpendicular, then: \[ \vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos 90^\circ \] \[ \vec{a}\cdot \vec{b}=0 \] So, B is true.

Step 3: Check statement C.
By Cauchy-Schwarz inequality: \[ |\vec{a}\cdot \vec{b}|\leq |\vec{a}||\vec{b}| \] So, C is true.

Step 4: Check statement D.
The cross product of two vectors is a vector, not a scalar. So, D is false.

Step 5: Check statement E.
The dot product of two vectors is a scalar. So, E is true. Thus, correct statements are: \[ A,\ B,\ C,\ E \] \[ \therefore \text{Correct Answer is (C)} \]