Class 12

Math

Algebra

Vector Algebra

Let $a=i^+4j^ +2k^,b=3i^−2j^ +7k^$and $c=2i^−j^ +4k^$. Find a vector $d$which is perpendicular to both $a$and $b$and $c$. $d=15$.

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If $xandy $ are two non-collinear vectors and $ABC$ isa triangle with side lengths $a,b,andc$ satisfying $(20a−15b)x+(15b−12c)y +(12c−20a)(× xy )=0,$ then triangle $ABC$ is a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. an isosceles triangle

Let $a,bandc$ be unit vectors, such that $a+b+c=x,ax˙=1,bx˙=23 ,∣x∣=2.$ Then find the angel between and $×˙$

Show that $(a−b)×(a+b)=2a×b$ and given a geometrical interpretation of it.

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 $cosθ=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}˙$

For given vector, $a$ = 2$i^$ $j$ +2$k^$ and $b$ = -$i^$ +$j^ $ - $k^$ , find the unit vector in the direction of the vector $a$ +$b$ .

If $A(−4,0,3)andB(14,2,−5),$ then which one of the following points lie on the bisector of the angle between $OAandOB(O$ is the origin of reference )? a. $(2,2,4)$ b. $(2,11,5)$ c. $(−3,−3,−6)$ d. $(1,1,2)$

In a triangle $PQR,SandT$ are points on $QRandPR,$ respectively, such that $QS=3SRandPT=4TR˙$ Let $M$ be the point of intersection of $PSandQT˙$ Determine the ratio $QM:MT$ using the vector method .

The median AD of the triangle ABC is bisected at E and BE meets AC at F. Find AF:FC.