87732 \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{x} \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}-\hat{\mathbf{k}}\). If the vector \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), then \(x=\)

87733 If \(\overrightarrow{\mathbf{a}}=\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}\)
and \(\overrightarrow{\mathbf{c}}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}\) are three vectors then which of the following statement is true.

87734 The value of \(\mathrm{m}\), if the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}-6 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+\mathrm{m} \hat{\mathbf{k}}\) are coplanar, is

87742 A unit vector perpendicular to both the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is

87732 \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{x} \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}-\hat{\mathbf{k}}\). If the vector \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), then \(x=\)

87733 If \(\overrightarrow{\mathbf{a}}=\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}\)
and \(\overrightarrow{\mathbf{c}}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}\) are three vectors then which of the following statement is true.

87734 The value of \(\mathrm{m}\), if the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}-6 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+\mathrm{m} \hat{\mathbf{k}}\) are coplanar, is

87742 A unit vector perpendicular to both the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is

87732 \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{x} \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}-\hat{\mathbf{k}}\). If the vector \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), then \(x=\)

87733 If \(\overrightarrow{\mathbf{a}}=\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}\)
and \(\overrightarrow{\mathbf{c}}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}\) are three vectors then which of the following statement is true.

87734 The value of \(\mathrm{m}\), if the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}-6 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+\mathrm{m} \hat{\mathbf{k}}\) are coplanar, is

87742 A unit vector perpendicular to both the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is

87732 \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\mathbf{x} \hat{\mathbf{i}}+(x-1) \hat{\mathbf{j}}-\hat{\mathbf{k}}\). If the vector \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), then \(x=\)

87733 If \(\overrightarrow{\mathbf{a}}=\mathbf{3} \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\mathbf{2} \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}\)
and \(\overrightarrow{\mathbf{c}}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}\) are three vectors then which of the following statement is true.

87734 The value of \(\mathrm{m}\), if the vectors \(\hat{\mathbf{i}}-\hat{\mathbf{j}}-6 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+\mathrm{m} \hat{\mathbf{k}}\) are coplanar, is

87742 A unit vector perpendicular to both the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is