Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

Find the value of λ so that the following vector is coplanar:

\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]

Find the value of λ so that the following vector is coplanar:

\[\vec{a} = 2 \hat{i} - \hat {j} + \hat {k} , \vec{b} = \hat {i} + 2 \hat {j} - 3 \hat {k} , \vec{c} = \lambda \hat {i} + \lambda \hat {j} + 5 \hat {k}\]

Find the value of λ so that the following vector is coplanar:

\[\vec{a} = \hat{i} + 2\hat { j} - 3 \hat {k} , \vec{b} = 3 \hat{i} + \lambda \hat {j} + \hat {k} , \vec{c} = \hat {i} + 2 \hat {j} + 2 \hat {k}\]

Find the value of λ so that the following vector is coplanar:

\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]

Show that four points whose position vectors are

\[6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}\]

Find the value of λ for which the four points with position vectors

\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]

Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]

\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{and} \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]

If c₁ = 1 and c₂ = 2, find c₃ which makes \[\vec{a,} \vec{b} \text { and } \vec{c}\] coplanar.

\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]

If c₂ = −1 and c₃ = 1, show that no value of c₁ can make \[\vec{a,} \vec{b}\text { and } \vec{c}\] coplanar.

If four points A, B, C and D with position vectors 4 \[\hat { i} +3\] \[\hat { j} +3\] \[\hat { k} ,5\] \[\hat { i} +\] \[x\hat { j} +7\] \[\hat { k} ,5\] \[\hat { i} +3\] \[\hat { j}\] and \[7 \hat{i} + 6 \hat{j} + \hat{k}\] respectively are coplanar, then find the value of x.

Write the value of \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]

Write the value of \[\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .\]

Write the value of \[\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .\]

Find the values of 'a' for which the vectors

\[\vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} , \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \text { and } \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat { k }\] are coplanar.

Find the volume of the parallelopiped with its edges represented by the vectors \[\hat {i} + \hat {j} , \hat {i} + 2 \hat {j} \text { and } \hat {i} + \hat {j} + \pi k .\]

If \[\vec{a,} \vec{b}\] \[\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .\]

If the vectors (sec² A) \[\hat {i} + \hat {j} + \hat {k} , \hat {i} + \left( \sec^2 B \right) \hat {j} + \hat {k} , \hat {i} + \hat {j} + \left( \sec^2 C \right) \hat {k}\] are coplanar, then find the value of cosec² A + cosec² B + cosec² C.