121211 If the vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}, 2 \hat{i}+\hat{j}-\hat{k}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2 \hat { \mathbf { k } }}\) are coplanar, then the value of \(\lambda\) is

121213 If the vectors \(a \hat{i}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}\) are coplanar \((a \neq b \neq c \neq 1)\), then the value of \(\mathbf{a b c}-(\mathbf{a}+\mathbf{b}+\mathbf{c})=\)

121214 The value of \([\vec{a}-\vec{b} \vec{b}-\vec{c} \vec{c}-\vec{a}]\) is equal to

121215 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}\) and \(\overrightarrow{\mathbf{r}}\) vectors defined by \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\), then the value of \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot q+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{r}}=\)

121216 If \(\vec{a}, \vec{b}\) and \(\overrightarrow{\mathbf{c}}\) are non-coplanar, then the value of \(\overrightarrow{\mathbf{a}} \cdot\left\{\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{\mathbf{3 \mathbf { b }} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})}\right\}-\overrightarrow{\mathbf{b}} \cdot\left\{\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{2 \overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})}\right\}\) is

121211 If the vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}, 2 \hat{i}+\hat{j}-\hat{k}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2 \hat { \mathbf { k } }}\) are coplanar, then the value of \(\lambda\) is

121213 If the vectors \(a \hat{i}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}\) are coplanar \((a \neq b \neq c \neq 1)\), then the value of \(\mathbf{a b c}-(\mathbf{a}+\mathbf{b}+\mathbf{c})=\)

121214 The value of \([\vec{a}-\vec{b} \vec{b}-\vec{c} \vec{c}-\vec{a}]\) is equal to

121215 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}\) and \(\overrightarrow{\mathbf{r}}\) vectors defined by \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\), then the value of \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot q+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{r}}=\)

121216 If \(\vec{a}, \vec{b}\) and \(\overrightarrow{\mathbf{c}}\) are non-coplanar, then the value of \(\overrightarrow{\mathbf{a}} \cdot\left\{\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{\mathbf{3 \mathbf { b }} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})}\right\}-\overrightarrow{\mathbf{b}} \cdot\left\{\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{2 \overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})}\right\}\) is

121211 If the vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}, 2 \hat{i}+\hat{j}-\hat{k}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2 \hat { \mathbf { k } }}\) are coplanar, then the value of \(\lambda\) is

121213 If the vectors \(a \hat{i}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}\) are coplanar \((a \neq b \neq c \neq 1)\), then the value of \(\mathbf{a b c}-(\mathbf{a}+\mathbf{b}+\mathbf{c})=\)

121214 The value of \([\vec{a}-\vec{b} \vec{b}-\vec{c} \vec{c}-\vec{a}]\) is equal to

121215 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}\) and \(\overrightarrow{\mathbf{r}}\) vectors defined by \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\), then the value of \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot q+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{r}}=\)

121216 If \(\vec{a}, \vec{b}\) and \(\overrightarrow{\mathbf{c}}\) are non-coplanar, then the value of \(\overrightarrow{\mathbf{a}} \cdot\left\{\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{\mathbf{3 \mathbf { b }} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})}\right\}-\overrightarrow{\mathbf{b}} \cdot\left\{\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{2 \overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})}\right\}\) is

121211 If the vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}, 2 \hat{i}+\hat{j}-\hat{k}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2 \hat { \mathbf { k } }}\) are coplanar, then the value of \(\lambda\) is

121213 If the vectors \(a \hat{i}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}\) are coplanar \((a \neq b \neq c \neq 1)\), then the value of \(\mathbf{a b c}-(\mathbf{a}+\mathbf{b}+\mathbf{c})=\)

121214 The value of \([\vec{a}-\vec{b} \vec{b}-\vec{c} \vec{c}-\vec{a}]\) is equal to

121215 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}\) and \(\overrightarrow{\mathbf{r}}\) vectors defined by \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\), then the value of \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot q+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{r}}=\)

121216 If \(\vec{a}, \vec{b}\) and \(\overrightarrow{\mathbf{c}}\) are non-coplanar, then the value of \(\overrightarrow{\mathbf{a}} \cdot\left\{\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{\mathbf{3 \mathbf { b }} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})}\right\}-\overrightarrow{\mathbf{b}} \cdot\left\{\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{2 \overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})}\right\}\) is

121211 If the vector \(2 \hat{i}-3 \hat{j}+4 \hat{k}, 2 \hat{i}+\hat{j}-\hat{k}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+\mathbf{2 \hat { \mathbf { k } }}\) are coplanar, then the value of \(\lambda\) is

121213 If the vectors \(a \hat{i}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}\) are coplanar \((a \neq b \neq c \neq 1)\), then the value of \(\mathbf{a b c}-(\mathbf{a}+\mathbf{b}+\mathbf{c})=\)

121214 The value of \([\vec{a}-\vec{b} \vec{b}-\vec{c} \vec{c}-\vec{a}]\) is equal to

121215 If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three non-coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}\) and \(\overrightarrow{\mathbf{r}}\) vectors defined by \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]} \cdot \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\), then the value of \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot q+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{r}}=\)

121216 If \(\vec{a}, \vec{b}\) and \(\overrightarrow{\mathbf{c}}\) are non-coplanar, then the value of \(\overrightarrow{\mathbf{a}} \cdot\left\{\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{\mathbf{3 \mathbf { b }} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})}\right\}-\overrightarrow{\mathbf{b}} \cdot\left\{\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{2 \overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})}\right\}\) is