Class 12

Math

Algebra

Vector Algebra

Write two different vectors having same direction.

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Let $D,EandF$ be the middle points of the sides $BC,CAandAB,$ respectively of a triangle $ABC˙$ Then prove that $AD+BE+CF=0$ .

Find a vector in the direction of the vector 5$i^$ - $j^ $ + 2$k^$ which has magnitude 8 units.

If $D,EandF$ are three points on the sides $BC,CAandAB,$ respectively, of a triangle $ABC$ such that the $CDBD ,AECE ,BFAF =−1$

If $b$ is a vector whose initial point divides thejoin of $5i^and5j^ $ in the ratio $k:1$ and whose terminal point is the origin and $∣∣ b∣∣ ≤37 ,thenk$ lies in the interval a. $[−6,−1/6]$ b. $(−∞,−6]∪[−1/6,∞)$ c. $[0,6]$ d. none of these

The position vector of the points $PandQ$ are $5i^+7j^ −2k^$ and $−3i^+3j^ +6k^$ , respectively. Vector $A=3i^−j^ +k^$ passes through point $P$ and vector $B=−3i^+2j^ +4k^$ passes through point $Q$ . A third vector $2i^+7j^ −5k^$ intersects vectors $AandB˙$ Find the position vectors of points of intersection.

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.

Show that the point $A,B$ and $C$ with position vectors $a$ =3$i^$ - 4$j^ $ -4$k^$ = 2$i^$ $j$ + $k^$ and $c$ = $i^$ - 3$j^ $ - 5$k^$ , respectively from the vertices of a right angled triangle.

Statement 1: $a=3i+pj +3k$ and $b=2i+3j +qk$ are parallel vectors if $p=9/2andq=2.$ Statement 2: if $a=a_{1}i+a_{2}j +a_{3}kandb=b_{1}i+b_{2}j +b_{3}k$ are parallel, then $b_{1}a_{1} =b_{2}a_{2} =b_{3}a_{3} ˙$