87744
If \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\), then a vector perpendicular to \(\vec{a}\) and in the plane containing \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) is
1 \(-17 \hat{i}+21 \hat{j}-97 \hat{k}\)
2 \(17 \hat{i}+21 \hat{j}-123 \hat{k}\)
3 \(-17 \hat{i}-21 \hat{j}+97 \hat{k}\)
4 \(-17 \hat{i}-21 \hat{j}-97 \hat{k}\)
Explanation:
(D) : Given, \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}\) \(\vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}\) \(\vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\) We know that, a vector perpendicular to a and in the plane containing \(\mathrm{b}\) and \(\mathrm{c}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc} \hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1 &2 &-5\\ 3 &5 &-1 \end{array}\right|\) \(=\hat{\mathrm{i}}(-2+25)-\hat{\mathrm{j}}(-1+15)+\hat{\mathrm{k}}(5-6)\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=23 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}} \\ 2& 3 &-1 \\ 23& -14& -1\end{array}\right|\) \(=\hat{\mathrm{i}}(-3-14)-\hat{\mathrm{j}}(-2+23)+\hat{\mathrm{k}}(-28-69)\) \( =-17 \hat{\mathrm{i}}-21 \hat{\mathrm{j}}-97 \hat{\mathrm{k}}\)
87744
If \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\), then a vector perpendicular to \(\vec{a}\) and in the plane containing \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) is
1 \(-17 \hat{i}+21 \hat{j}-97 \hat{k}\)
2 \(17 \hat{i}+21 \hat{j}-123 \hat{k}\)
3 \(-17 \hat{i}-21 \hat{j}+97 \hat{k}\)
4 \(-17 \hat{i}-21 \hat{j}-97 \hat{k}\)
Explanation:
(D) : Given, \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}\) \(\vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}\) \(\vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\) We know that, a vector perpendicular to a and in the plane containing \(\mathrm{b}\) and \(\mathrm{c}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc} \hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1 &2 &-5\\ 3 &5 &-1 \end{array}\right|\) \(=\hat{\mathrm{i}}(-2+25)-\hat{\mathrm{j}}(-1+15)+\hat{\mathrm{k}}(5-6)\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=23 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}} \\ 2& 3 &-1 \\ 23& -14& -1\end{array}\right|\) \(=\hat{\mathrm{i}}(-3-14)-\hat{\mathrm{j}}(-2+23)+\hat{\mathrm{k}}(-28-69)\) \( =-17 \hat{\mathrm{i}}-21 \hat{\mathrm{j}}-97 \hat{\mathrm{k}}\)
87744
If \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\), then a vector perpendicular to \(\vec{a}\) and in the plane containing \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) is
1 \(-17 \hat{i}+21 \hat{j}-97 \hat{k}\)
2 \(17 \hat{i}+21 \hat{j}-123 \hat{k}\)
3 \(-17 \hat{i}-21 \hat{j}+97 \hat{k}\)
4 \(-17 \hat{i}-21 \hat{j}-97 \hat{k}\)
Explanation:
(D) : Given, \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}\) \(\vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}\) \(\vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\) We know that, a vector perpendicular to a and in the plane containing \(\mathrm{b}\) and \(\mathrm{c}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc} \hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1 &2 &-5\\ 3 &5 &-1 \end{array}\right|\) \(=\hat{\mathrm{i}}(-2+25)-\hat{\mathrm{j}}(-1+15)+\hat{\mathrm{k}}(5-6)\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=23 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}} \\ 2& 3 &-1 \\ 23& -14& -1\end{array}\right|\) \(=\hat{\mathrm{i}}(-3-14)-\hat{\mathrm{j}}(-2+23)+\hat{\mathrm{k}}(-28-69)\) \( =-17 \hat{\mathrm{i}}-21 \hat{\mathrm{j}}-97 \hat{\mathrm{k}}\)
87744
If \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\), then a vector perpendicular to \(\vec{a}\) and in the plane containing \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) is
1 \(-17 \hat{i}+21 \hat{j}-97 \hat{k}\)
2 \(17 \hat{i}+21 \hat{j}-123 \hat{k}\)
3 \(-17 \hat{i}-21 \hat{j}+97 \hat{k}\)
4 \(-17 \hat{i}-21 \hat{j}-97 \hat{k}\)
Explanation:
(D) : Given, \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}\) \(\vec{b}=\hat{i}+2 \hat{j}-5 \hat{k}\) \(\vec{c}=3 \hat{i}+5 \hat{j}-\hat{k}\) We know that, a vector perpendicular to a and in the plane containing \(\mathrm{b}\) and \(\mathrm{c}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc} \hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1 &2 &-5\\ 3 &5 &-1 \end{array}\right|\) \(=\hat{\mathrm{i}}(-2+25)-\hat{\mathrm{j}}(-1+15)+\hat{\mathrm{k}}(5-6)\) \(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=23 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}} \\ 2& 3 &-1 \\ 23& -14& -1\end{array}\right|\) \(=\hat{\mathrm{i}}(-3-14)-\hat{\mathrm{j}}(-2+23)+\hat{\mathrm{k}}(-28-69)\) \( =-17 \hat{\mathrm{i}}-21 \hat{\mathrm{j}}-97 \hat{\mathrm{k}}\)
