In Figure, which of the vectors are: (i) Collinear (ii) Equal (iii) Coinitial
Prove that the necessary and sufficient condition for any four points in three-dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.
In a four-dimensional space where unit vectors along the axes are i^,j^,k^andl^,anda1,a2,a3,a4 are four non-zero vectors such that no vector can be expressed as a linear combination of others and (λ−1)(a1−a2)+μ(a2+a3)+γ(a3+a4−2a2)+a3+δa4=0, then a. λ=1 b. μ=−2/3 c. γ=2/3 d. δ=1/3
If the vectors 3p+q;5p−3qand2p+q;4p−2q are pairs of mutually perpendicular vectors, then find the angle between vectors pandq˙
Statement 1:Let A(a),B(b)andC(c) be three points such that a=2i^+k^,b=3i^−j^+3k^andc=−i^+7j^−5k^˙ Then OABC is a tetrahedron. Statement 2: Let A(a),B(b)andC(c) be three points such that vectors a,bandc are non-coplanar. Then OABC is a tetrahedron where O is the origin.
A isa vector with direction cosines cosα,cosβandcosγ˙ Assuming the y−z plane as a mirror, the directin cosines of the reflected image of A in the plane are a. cosα,cosβ,cosγ b. cosα,−cosβ,cosγ c. −cosα,cosβ,cosγ d. −cosα,−cosβ,−cosγ