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JEE MATHEMATICS MCQ: ADDITION OF VECTORS

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If the vectors r1→=ai^+j^+k^,r2→=i^+bj^+k^,r3→=i^+j^+ck^(a≠1,b≠1,c≠1)\overrightarrow {{r_1}} = a\hat i + \hat j + \hat k,\overrightarrow {{r_2}} = \hat i + b\hat j + \hat k,\overrightarrow {{r_3}} = \hat i + \hat j + c\hat k\left( {a \ne 1,b \ne 1,c \ne 1} \right) are coplanar, then the value of 11−a+11−b+11−c,\frac{1}{{1 - a}} + \frac{1}{{1 - b}} + \frac{1}{{1 - c}}, is
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Correct option: C
1
Option C (1) is correct because the coplanarity condition leads to an equation that simplifies the given expression to 1.

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Let a=i^−j^,b=j^−k^\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}} and c=k^−i^\mathbf{c}=\hat{\mathbf{k}}-\hat{\mathbf{i}}. If d\mathbf{d} is aa unit vector such that a⋅d=0=(bc\mathbf{a} \cdot \mathbf{d}=0=(b \mathbf{c} ) , then d\mathbf{d} is equal to
 A vector a⃗{\mathbf{\vec a}} has components 2  p2\;p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense, if this respect to new system a⃗{\mathbf{\vec a}} has components p+1p + 1 and 1, then If a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are three mutually perpendicular unit vetors and d⃗\vec{d} is a unit vector which makes equal angles with a→,b→\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} ānd c→\overrightarrow{\mathrm{c}} then the value of ∣a→+b→+c→+d→∣2|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|^2 The non-zero vectors are a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are related by a⃗=8b⃗\vec{a}=8 \vec{b} and c⃗=−7b⃗\vec{c}=-7 \vec{b}. Then the angle between a⃗\vec{a} and c⃗\vec{c} is If the vectors 2i+j−k,−i+2j+λk2 \mathbf{i}+\mathbf{j}-\mathbf{k},-\mathbf{i}+2 \mathbf{j}+\lambda \mathbf{k} and −5i+2j−k-5 \mathbf{i}+2 \mathbf{j}-\mathbf{k} ar coplanar, then the value of λ\lambda is equal The medium ADAD of the triangle ABCABC is bisected at E,  BEE,\;BE meets ACAC in FF, then AF:AC=AF:AC =
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