88185 If \(a, b, c\) are non-zero, non-collinear vectors and \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\), then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\)

88186 If the vectors \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}\) (where, \(\mathbf{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1\) ) are coplanar, then the value of \(p q r-(p+q+r)\) is

88187 The sum of the distinct real values of \(\mu\) for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\) are coplanar, is

88188 Let \(a=\hat{i}+2 \hat{j}+4 \hat{k}, b=\hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(\mathbf{c}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}+\left(\lambda^2-1\right) \hat{\mathbf{k}}\) be coplanar vectors. Then, the non-zero vector \(\mathbf{a} \times \mathbf{c}\) is

88189 Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},\)
\((1+b) \hat{i}+2 b \hat{j}-b \hat{k} \text { and }(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k},\)
\(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbf{R}\) be coplanar.
Then, which of the following is true?

88185 If \(a, b, c\) are non-zero, non-collinear vectors and \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\), then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\)

88186 If the vectors \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}\) (where, \(\mathbf{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1\) ) are coplanar, then the value of \(p q r-(p+q+r)\) is

88187 The sum of the distinct real values of \(\mu\) for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\) are coplanar, is

88188 Let \(a=\hat{i}+2 \hat{j}+4 \hat{k}, b=\hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(\mathbf{c}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}+\left(\lambda^2-1\right) \hat{\mathbf{k}}\) be coplanar vectors. Then, the non-zero vector \(\mathbf{a} \times \mathbf{c}\) is

88189 Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},\)
\((1+b) \hat{i}+2 b \hat{j}-b \hat{k} \text { and }(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k},\)
\(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbf{R}\) be coplanar.
Then, which of the following is true?

88185 If \(a, b, c\) are non-zero, non-collinear vectors and \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\), then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\)

88186 If the vectors \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}\) (where, \(\mathbf{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1\) ) are coplanar, then the value of \(p q r-(p+q+r)\) is

88187 The sum of the distinct real values of \(\mu\) for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\) are coplanar, is

88188 Let \(a=\hat{i}+2 \hat{j}+4 \hat{k}, b=\hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(\mathbf{c}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}+\left(\lambda^2-1\right) \hat{\mathbf{k}}\) be coplanar vectors. Then, the non-zero vector \(\mathbf{a} \times \mathbf{c}\) is

88189 Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},\)
\((1+b) \hat{i}+2 b \hat{j}-b \hat{k} \text { and }(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k},\)
\(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbf{R}\) be coplanar.
Then, which of the following is true?

88185 If \(a, b, c\) are non-zero, non-collinear vectors and \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\), then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\)

88186 If the vectors \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}\) (where, \(\mathbf{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1\) ) are coplanar, then the value of \(p q r-(p+q+r)\) is

88187 The sum of the distinct real values of \(\mu\) for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\) are coplanar, is

88188 Let \(a=\hat{i}+2 \hat{j}+4 \hat{k}, b=\hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(\mathbf{c}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}+\left(\lambda^2-1\right) \hat{\mathbf{k}}\) be coplanar vectors. Then, the non-zero vector \(\mathbf{a} \times \mathbf{c}\) is

88189 Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},\)
\((1+b) \hat{i}+2 b \hat{j}-b \hat{k} \text { and }(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k},\)
\(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbf{R}\) be coplanar.
Then, which of the following is true?

88185 If \(a, b, c\) are non-zero, non-collinear vectors and \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\), then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\)

88186 If the vectors \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}\) (where, \(\mathbf{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1\) ) are coplanar, then the value of \(p q r-(p+q+r)\) is

88187 The sum of the distinct real values of \(\mu\) for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\) are coplanar, is

88188 Let \(a=\hat{i}+2 \hat{j}+4 \hat{k}, b=\hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(\mathbf{c}=\mathbf{2} \hat{\mathbf{i}}+\mathbf{4} \hat{\mathbf{j}}+\left(\lambda^2-1\right) \hat{\mathbf{k}}\) be coplanar vectors. Then, the non-zero vector \(\mathbf{a} \times \mathbf{c}\) is

88189 Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},\)
\((1+b) \hat{i}+2 b \hat{j}-b \hat{k} \text { and }(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k},\)
\(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbf{R}\) be coplanar.
Then, which of the following is true?