Show that the points A, B and C with position vectors, a=3i^−4j^−4k^, b=2i^−j^+k^and c=i^−3j^−5k^ respectively form the vertices of a right angled triangle.
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˙
Let x2+3y2=3 be the equation of an ellipse in the x−y plane. AandB are two points whose position vectors are −3i^and−3i^+2k^˙ Then the position vector of a point P on the ellipse such that ∠APB=π/4 is a. ±j^ b. ±(i^+j^) c. ±i^ d. none of these
Given three points are A(−3,−2,0),B(3,−3,1)andC(5,0,2)˙ Then find a vector having the same direction as that of AB and magnitude equal to ∣∣AC∣∣˙
A unit vector of modulus 2 is equally inclined to x - and y -axes at an angle π/3 . Find the length of projection of the vector on the z -axis.
If the projections of vector a on x -, y - and z -axes are 2, 1 and 2 units ,respectively, find the angle at which vector a is inclined to the z -axis.
If a,bandc are any three non-coplanar vectors, then prove that points l1a+m1b+n1c,l2a+m2b+n2c,l3a+m3b+n3c,l4a+m4b+n4c are coplanar if ⎣⎡l1m1n11l2m2n21l3m3n31l4m4n41⎦⎤=0