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  • Provide solution for RD Sharma Maths Class 12 Chapter 25 Scalar Triple Product Exercise Very Short Answer Question, question 8.

Provide solution for RD Sharma Maths Class 12 Chapter 25 Scalar Triple Product Exercise Very Short Answer Question, question 8.

Answers (1)

Answer:

144

Hint:

Use scalar triple product formula.

Given:

 |\vec{a}|=3 \text { and }|\vec{b}|=4 \text { and }\left[\begin{array}{lll} \overrightarrow{\boldsymbol{a}} & \vec{b} & \vec{a} \times \vec{b}]+(\vec{a} \cdot \vec{b})^{2} \end{array}\right.

Solution:

\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{a} \times \vec{b} \end{array}\right]+(\vec{a} \cdot \vec{b})^{2}

\begin{aligned} &=\{(\vec{a}) \times(\vec{b}) \cdot(\vec{a} \times \vec{b})\}+(\vec{a} \cdot \vec{b})^{2}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c}] \end{array}=(\vec{a} \times \vec{b}) \cdot \vec{c}\right]\right.\\ &=(\vec{a} \times \vec{b}) \cdot(\vec{a} \times \vec{b})+(\vec{a} \cdot \vec{b})^{2} \end{aligned}

\begin{aligned} &=|(\vec{a} \times \vec{b})|^{2}+(\vec{a} \cdot \vec{b})^{2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because \vec{a} \times \vec{a}=|\vec{a}|^{2}\right] \\ &=(|\vec{a}||\vec{b}| \sin \theta)^{2}+(|\vec{\alpha}||\vec{b}| \cos \theta)^{2} & {\left[\begin{array}{l} \because \vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta \\ \vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \end{array}\right]} \end{aligned}

\begin{aligned} &=(|\vec{a}||\vec{b}|)^{2} \sin ^{2} \theta+(|\vec{a}||\vec{b}|)^{2} \cos ^{2} \theta \\ &=(|\vec{a}||\vec{b}|)^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right) \\ &=(|\vec{a}||\vec{b}|)^{2} \cdot 1 \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because \sin ^{2} A+\cos ^{2} A=1\right] \\ &=(3.4)^{2}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad[\because|\vec{a}|=3,|\vec{b}|=4] \\ &=(12)^{2}=144 \end{aligned}

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