Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (5,7).
i. Prove that the points a−2b+3c,2a+3b−4cand−7b+10c are are collinear, where a,b,c are non-coplanar. ii. Prove that the points A(1,2,3),B(3,4,7),andC(−3,−2,−5) are collinear. find the ratio in which point C divides AB.
If a,b are two non-collinear vectors, prove that the points with position vectors a+b,a−b and a+λb are collinear for all real values of λ˙
The number of distinct real values of λ , for which the vectors λ2i^+j^+k,i^−λ2j^+k^andi^+j^−λ2k^ are coplanar is a. zero b. one c. two d. three
If A,B,C,D are four distinct point in space such that AB is not perpendicular to CD and satisfies ABC˙D=k(∣∣AD∣∣2+∣∣BC∣∣2−∣∣AC∣∣2−∣∣BD∣∣2), then find the value of k˙
Let us define the length of a vector ai^+bj^+ck^as∣a∣+∣b∣+∣c∣˙ This definition coincides with the usual definition of length of a vector ai^+bj^+ck^ is and only if a. a=b=c=0 b. any two of a,b,andc are zero c. any one of a,b,andc is zero d. a+b+c=0
Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l1,m1,n1andl2,m2,n2 are proportional to l1+l2,m1+m2,n1+n2˙ Statement 2: The angle between the two intersection lines having direction cosines as l1,m1,n1andl2,m2,n2 is given by cosθ=l1l2+m1m2+n1n2˙