Find the volume of the parallelepiped whose edges are represented by the vectors $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})_{\text {and }} \overrightarrow{\mathrm{c}}=(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

Rs aggarwal Class 12 Chapter Product of three vectors Exercise 25 B

Solution

$\begin{array}{l} \vec{a}=2 \hat{\imath}-3 \hat{\jmath}+4 k \\ \vec{b}=\hat{\imath}+2 \hat{\jmath}-k \\ \vec{c}=3 \hat{\imath}-2 \hat{\jmath}+2 k \end{array}$

To find the volume of parallelepiped it is sufficient to find $[\vec{a} \vec{b} \vec{c}]$

$\vec{a} \cdot \vec{b} \times \vec{c}=\left[\begin{array}{llll} \vec{a} & \vec{b} & \vec{c} \end{array}\right]=\begin{array}{ccc} 2 & -3 & 4 \\ & 2 & -10 \\ & 3 & -2 & 2 \end{array}$

But volume cannot be negative

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$\text { de direction cosines of the vector } \overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$

$\begin{array}{l} \text { (d) } \vec{P}=\hat{\imath}+5 \hat{\jmath}+4 \hat{k}\\ \widehat{D} \vec{Q}=4 \hat{\imath}+\hat{\jmath}-2 \hat{k}\\ \text { BH) } \mid \hat{0}=3 \hat{\imath}-4 \hat{\jmath}-6 \hat{k}\\ \text { Direction ratios of } \forall H) \overrightarrow{\mathbb{Q}} \text { are }(3,-4,-6) \end{array}$

Find the direction cosines of a vector which is equally inclined to the $x-a x i s, y-a x i s$ and $z$ – axis.

$\text { When does }|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|_{\text {hold? }}$

$\text { If } \theta \text { is the angle between } \overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}, \text { and }|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|_{\text {then what is the value of } \theta \text { ? }}$

$\text { If } \overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+27 \hat{\mathrm{k}})_{\text {and }} \overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\mu \hat{\mathrm{k}})_{\text {are such that }} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0} \text { then find the values }$
of $\lambda$ and $\mu$ .

Write the value of
$\hat{\mathrm{i}} \cdot(\hat{\mathrm{j}} \times \hat{\mathrm{k}})+\hat{\mathrm{j}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{k}})+\hat{\mathrm{k}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{j}})$

$\text { en the vectors } \overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+3 \hat{\mathrm{k}})_{\text {and }} \overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+9 \hat{\mathrm{k}})_{\text {are parallel. }}$

What conclusion can you draw about vectors $\vec{a}$ and $\vec{b}$ when $\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{b}=0 ?$

Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes 1 and 2 respectively, when $|\vec{a} \times \vec{b}|=\sqrt{3}$

$\text { If } \vec{a}=(\hat{i}-7 \hat{j}+7 \hat{k})_{\text {and }} \vec{b}=(3 \hat{i}-2 \hat{j}+2 \hat{k})_{\text {then find }}|\vec{a} \times \vec{b}|$

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More Material

$\text { de direction cosines of the vector } \overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$

$\text { de direction cosines of the vector } \overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$

Find the direction cosines of a vector which is equally inclined to the $x-a x i s, y-a x i s$ and $z$ – axis.

$\text { When does }|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|_{\text {hold? }}$

Write the value of
$\hat{\mathrm{i}} \cdot(\hat{\mathrm{j}} \times \hat{\mathrm{k}})+\hat{\mathrm{j}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{k}})+\hat{\mathrm{k}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{j}})$

$\text { en the vectors } \overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+3 \hat{\mathrm{k}})_{\text {and }} \overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+9 \hat{\mathrm{k}})_{\text {are parallel. }}$

What conclusion can you draw about vectors $\vec{a}$ and $\vec{b}$ when $\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{b}=0 ?$

Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes 1 and 2 respectively, when $|\vec{a} \times \vec{b}|=\sqrt{3}$

$\text { If } \vec{a}=(\hat{i}-7 \hat{j}+7 \hat{k})_{\text {and }} \vec{b}=(3 \hat{i}-2 \hat{j}+2 \hat{k})_{\text {then find }}|\vec{a} \times \vec{b}|$

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Find the direction cosines of a vector which is equally inclined to the and – axis.

of and .

Write the value of

What conclusion can you draw about vectors and when and

Find the angle between two vectors and with magnitudes 1 and 2 respectively, when

Find the direction cosines of a vector which is equally inclined to the $x-a x i s, y-a x i s$ and $z$ – axis.

Write the value of
$\hat{\mathrm{i}} \cdot(\hat{\mathrm{j}} \times \hat{\mathrm{k}})+\hat{\mathrm{j}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{k}})+\hat{\mathrm{k}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{j}})$

What conclusion can you draw about vectors $\vec{a}$ and $\vec{b}$ when $\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{b}=0 ?$

Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes 1 and 2 respectively, when $|\vec{a} \times \vec{b}|=\sqrt{3}$