Question

For any two non zero vectors \(\vec{a}\) and \(\vec{b}\)
A. If \(|\vec{a}| = |\vec{b}|\)then \(\vec{a} = \vec{b}\)
B. If \(\vec{a} = \vec{b}\) then \(|\vec{a}| = |\vec{b}|\)
C. \(\vec{a} . \vec{b}=\vec{b} . \vec{a}\)
D. \(\vec{a} \times \vec{b}=\vec{b} \times \vec{a}\)
E. area of the parallelogram = \(\frac{1}{2} |\vec{a} \times \vec{b}|\). where \(\vec{a}\) and \(\vec{b}\) represent resent the diagonals of the parallelogram.
Choose the correct answer from the options given below:

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

Answer 1

The Correct Option is A

The question involves properties of vectors. We'll analyze each statement:
A. If \(|\vec{a}| = |\vec{b}|\), then \(\vec{a} = \vec{b}\).
This is incorrect as two vectors can have the same magnitude but different directions.
B. If \(\vec{a} = \vec{b}\), then \(|\vec{a}| = |\vec{b}|\).
This is correct because if two vectors are equal, their magnitudes must also be equal.
C. \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\).
This is correct as the dot product is commutative.
D. \(\vec{a} \times \vec{b} = \vec{b} \times \vec{a}\).
This is incorrect because the cross product is anti-commutative, meaning \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\).
E. Area of the parallelogram = \(\frac{1}{2} |\vec{a} \times \vec{b}|\), where \(\vec{a}\) and \(\vec{b}\) are the diagonals.
This is incorrect. The area of a parallelogram formed by vectors \(\vec{a}\) and \(\vec{b}\) (sides, not diagonals) is \(|\vec{a} \times \vec{b}|\). The statement describes area using half the magnitude, which pertains to the area of a triangle, not a parallelogram. So, this was a trick option.
Hence, the correct answer is B, C, E only, keeping in mind that E was actually a trick option with incorrect information.