88174 If the vectors \(\vec{a}+\lambda \vec{b}+3 \vec{c},-2 \vec{a}+3 \vec{b}-4 \vec{c}\) and \(\vec{a}-3 \vec{b}+\mathbf{5} \vec{c}\) are coplanar, then the value of \(\lambda\) is

88182 If \(a=\alpha i+3 j-6 k\) and \(b=2 i-j+\beta k\), then the values of \(\alpha, \beta\) so that \(a\) and \(b\) may be collinear are

88172 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}\) are coplanar, then \(\lambda=\)

88173 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\) are linearly dependent vectors and \(|\overrightarrow{\mathbf{c}}|=\sqrt{3}\), then

88175 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\mathbf{y i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

88174 If the vectors \(\vec{a}+\lambda \vec{b}+3 \vec{c},-2 \vec{a}+3 \vec{b}-4 \vec{c}\) and \(\vec{a}-3 \vec{b}+\mathbf{5} \vec{c}\) are coplanar, then the value of \(\lambda\) is

88182 If \(a=\alpha i+3 j-6 k\) and \(b=2 i-j+\beta k\), then the values of \(\alpha, \beta\) so that \(a\) and \(b\) may be collinear are

88172 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}\) are coplanar, then \(\lambda=\)

88173 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\) are linearly dependent vectors and \(|\overrightarrow{\mathbf{c}}|=\sqrt{3}\), then

88175 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\mathbf{y i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

88174 If the vectors \(\vec{a}+\lambda \vec{b}+3 \vec{c},-2 \vec{a}+3 \vec{b}-4 \vec{c}\) and \(\vec{a}-3 \vec{b}+\mathbf{5} \vec{c}\) are coplanar, then the value of \(\lambda\) is

88182 If \(a=\alpha i+3 j-6 k\) and \(b=2 i-j+\beta k\), then the values of \(\alpha, \beta\) so that \(a\) and \(b\) may be collinear are

88172 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}\) are coplanar, then \(\lambda=\)

88173 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\) are linearly dependent vectors and \(|\overrightarrow{\mathbf{c}}|=\sqrt{3}\), then

88175 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\mathbf{y i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

88174 If the vectors \(\vec{a}+\lambda \vec{b}+3 \vec{c},-2 \vec{a}+3 \vec{b}-4 \vec{c}\) and \(\vec{a}-3 \vec{b}+\mathbf{5} \vec{c}\) are coplanar, then the value of \(\lambda\) is

88182 If \(a=\alpha i+3 j-6 k\) and \(b=2 i-j+\beta k\), then the values of \(\alpha, \beta\) so that \(a\) and \(b\) may be collinear are

88172 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}\) are coplanar, then \(\lambda=\)

88173 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\) are linearly dependent vectors and \(|\overrightarrow{\mathbf{c}}|=\sqrt{3}\), then

88175 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\mathbf{y i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

88174 If the vectors \(\vec{a}+\lambda \vec{b}+3 \vec{c},-2 \vec{a}+3 \vec{b}-4 \vec{c}\) and \(\vec{a}-3 \vec{b}+\mathbf{5} \vec{c}\) are coplanar, then the value of \(\lambda\) is

88182 If \(a=\alpha i+3 j-6 k\) and \(b=2 i-j+\beta k\), then the values of \(\alpha, \beta\) so that \(a\) and \(b\) may be collinear are

88172 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{c}}=\mathbf{2} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}\) are coplanar, then \(\lambda=\)

88173 If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\) are linearly dependent vectors and \(|\overrightarrow{\mathbf{c}}|=\sqrt{3}\), then

88175 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\mathbf{y i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on