88001 If \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}\), then find \((\vec{a}+3 \vec{b}) \cdot(2 \vec{a}-\vec{b})\).

88002 The value of \([\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}]\) where \(|\overrightarrow{\mathbf{a}}|=1\), \(|\overrightarrow{\mathrm{b}}|=5,|\overrightarrow{\mathrm{c}}|=3\), is

88003 Angle between the vectors \(\sqrt{3}(\vec{a} \times \vec{b})\) and \(\overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}\) is

88005 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

88001 If \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}\), then find \((\vec{a}+3 \vec{b}) \cdot(2 \vec{a}-\vec{b})\).

88002 The value of \([\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}]\) where \(|\overrightarrow{\mathbf{a}}|=1\), \(|\overrightarrow{\mathrm{b}}|=5,|\overrightarrow{\mathrm{c}}|=3\), is

88003 Angle between the vectors \(\sqrt{3}(\vec{a} \times \vec{b})\) and \(\overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}\) is

88005 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

88001 If \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}\), then find \((\vec{a}+3 \vec{b}) \cdot(2 \vec{a}-\vec{b})\).

88002 The value of \([\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}]\) where \(|\overrightarrow{\mathbf{a}}|=1\), \(|\overrightarrow{\mathrm{b}}|=5,|\overrightarrow{\mathrm{c}}|=3\), is

88003 Angle between the vectors \(\sqrt{3}(\vec{a} \times \vec{b})\) and \(\overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}\) is

88005 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

88001 If \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}\), then find \((\vec{a}+3 \vec{b}) \cdot(2 \vec{a}-\vec{b})\).

88002 The value of \([\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}]\) where \(|\overrightarrow{\mathbf{a}}|=1\), \(|\overrightarrow{\mathrm{b}}|=5,|\overrightarrow{\mathrm{c}}|=3\), is

88003 Angle between the vectors \(\sqrt{3}(\vec{a} \times \vec{b})\) and \(\overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}\) is

88005 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is