SATHEE: Vectors 3 Question 18

Vectors 3 Question 18

18. The scalar $\vec{A} \cdot [(\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C})]$ equals

(1981, 2M)

(a) 0

(b) $[\vec{A} \vec{B} \vec{C}] + [\begin{array}{lll} B & \vec{C} & \vec{A} \end{array}]$

(d) None of the above

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Answer:

Correct Answer: 18. (a)

Solution:

$\vec{A} \cdot (\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C})$

$[∵$ it is a scalar triple product of three vectors of the form $\vec{A}, \vec{B} + \vec{C}, \vec{A} + \vec{B} + \vec{C}]$

$\begin{aligned} = \vec{A} \cdot (\vec{B} \times \vec{A} + \vec{B} \times \vec{B} \\ + \vec{B} \times \vec{C} + \vec{C} \times \vec{A} + \vec{C} \times \vec{B} + \vec{C} \times \vec{C}) \\ = \vec{A} \cdot (\vec{B} \times \vec{A}) + \vec{A} \cdot (\vec{B} \times \vec{C}) + \vec{A} \cdot (\vec{C} \times \vec{B}) \\ = [\vec{A} \vec{B} \vec{A}] - [\vec{A} \vec{B} \vec{C}] = 0 \end{aligned}$

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Vectors 3 Question 18

18. The scalar A→⋅[(B→+C→)×(A→+B→+C→)] equals

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18. The scalar $\vec{A} \cdot [(\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C})]$ equals