Class 12

Math

Algebra

Vector Algebra

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

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If $r_{1},r_{2},r_{3}$ are the position vectors of the collinear points and scalar $pandq$ exist such that $r_{3}=pr_{1}+qr_{2},$ then show that $p+q=1.$

Let $a,b,andcanda_{_{′}},b_{_{′}},c_{′}$ are reciprocal system of vectors, then prove that $a_{_{′}}×b_{_{′}}+b_{_{′}}×c_{_{′}}+c_{_{′}}×a_{_{′}}=[abc]a+b+c $ .

Find the least positive integral value of $x$ for which the angel between vectors $a=xi^−3j^ −k^$ and $b=2xi^+xj^ −k^$ is acute.

If 2$AC$ = 3$CB$ , then prove that 2$OA$ =3$CB$ then prove that 2$OA$ + 3$OB$ =5$OC$ where $O$ is the origin.

If $a=7i^−4k^andb=−2i^−j^ +2k^,$ determine vector $c$ along the internal bisector of the angle between of the angle between vectors $aandbsuchthat∣c∣$ =5$6 $

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 $∣∣ AC∣∣ ˙$

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 $ABC$ be triangle, the position vecrtors of whose vertices are respectively $i^+2j^ +4k^$ , -2$i^+2j^ +k^and2i^+4j^ −3k^$ . Then $DeltaABC$ is a. isosceles b. equilateral c. right angled d. none of these