Class 12

Math

Algebra

Vector Algebra

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.

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Find the angle of vector $a$ = 6$i^$ + 2$j^ $ - 3$k^$ with $x$ -axis.

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ are proportional to $l_{1}+l_{2},m_{1}+m_{2},n_{1}+n_{2}˙$ Statement 2: The angle between the two intersection lines having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ is given by $cosθ=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}˙$

Let $x_{2}+3y_{2}=3$ be the equation of an ellipse in the $x−y$ plane. $AandB$ are two points whose position vectors are $−3 i^and−3 i^+2k^˙$ 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 $l_{1}a+m_{1}b+n_{1}c,l_{2}a+m_{2}b+n_{2}c,l_{3}a+m_{3}b+n_{3}c,l_{4}a+m_{4}b+n_{4}c$ are coplanar if $⎣⎡ l_{1}m_{1}n_{1}1 l_{2}m_{2}n_{2}1 l_{3}m_{3}n_{3}1 l_{4}m_{4}n_{4}1 ⎦⎤ =0$

Column I, Column II Collinear vectors, p.$a$ Coinitial vectors, q. $b$ Equal vectors, r. $c$ Unlike vectors (same intitial point), s. $d$