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JEE MATHEMATICS MCQ: ADDITION OF VECTORS

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Let the vectors a⃗=(1+t)i^+(1−t)j^+k^\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, b⃗=(1−t)i^+(1+t)j^+2k^\vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k} and c⃗=ti^−tj^+k^,t∈R\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R} be such that for α,β,γ∈R,αa⃗+βb⃗+γc⃗=0→\alpha, \beta, \gamma \in \mathbf{R}, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} ⇒α=β=γ=0\Rightarrow \alpha=\beta=\gamma=0. Then, the set of all values of tt is
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Correct option: C
Equal to R−{0}\mathbf{R}-\{0\}
Option D (Equal to R−{0}\mathbf{R}-\{0\}) is correct because the determinant of the matrix formed by the vectors is non-zero for all tt except t=0t=0, ensuring the vectors are linearly independent.

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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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