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JEE MATHEMATICS MCQ: ADDITION OF VECTORS

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Application +4 | -1
On the xy plane where O is the origin, given points, A(1,0),  B(0,1)   and   C(1,1).A(1,0),\,\,B(0,1){\text{ }}\,\,{\text{and}}\,\,{\text{ }}C(1,1). Let P, Q and R be moving point on the line OA,  OB,  OCOA,\,\,OB,\,\,OC respectively such that OP=45t(OA),    OQ=60t(OB),    OR=(1−t)(OC)OP = 45t(OA),\,\,\,\,OQ = 60t(OB),\,\,\,\,OR = (1 - t)(OC) with t>0.t > 0. If the three points P, Q and R are collinear, then the value of t is equal to
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Correct option: B
7/187
Option B (7/187) is correct because it is the value of t that makes the points P, Q, and R collinear, derived by equating the slopes of PQ and QR.

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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
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