Find a vector d that is perpendicular to both b and c, and satisfies d·a=21, given a=4i+5j-k, b=i-4j+5k, c=3i+j-k. Solved by setting up and solving a linear system.
JEE Main 2023 PYQ: triangle PQR with A,B,C on its sides dividing them in ratio 1:2. Find Area(PQR)/Area(ABC) using position vectors and the cross-product area formula.
A vector proof of the midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.
Prove P, Q, R are collinear given vector PO + OQ = vector QO + OR, by writing everything in position vectors and showing Q is the midpoint of P and R.
Arc AC subtends 90° at centre O; point B divides it so arc AB:arc BC=1:5. Given OC=αOA+βOB, find α+√2(√3−1)β by placing the points using their subtended angles and comparing vector components.
Prove the vector triangle inequality |a+b|≤|a|+|b| using the dot product expansion of |a+b|² and the Cauchy-Schwarz bound a·b≤|a||b|.
Given x, y, z are unit vectors in 3D space, find the minimum value of |x+y|²+|y+z|²+|z+x|² by relating the sum to |x+y+z|² and using that a squared magnitude can’t be negative.
Find the greatest value of √3|a+b|+|a−b| where a and b are unit vectors, by writing both magnitudes in terms of the half-angle between them.
Given a=i−j+2k, b=2i+4j+k, and c=λi+j+μk are mutually orthogonal, find (λ,μ) by setting the dot products a·c and b·c to zero.
For p>0, vector v2=2i+(p+1)j is obtained by rotating v1=√3pi+j by angle θ. Given tanθ=(α√3−2)/(4√3+3), find α using the rotation-preserves-length property.