Class 12

Math

Algebra

Vector Algebra

If the resultant of three forces $F_{1}=pi^+3j^ −k^,F_{2}=6i^−k^andF_{3}=−5i^+j^ +2k^$ acting on a parricle has magnitude equal to 5 units, then the value of $p$ is a. $−6$ b. $−4$ c. $2$ d. $4$

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

Prove that the necessary and sufficient condition for any four points in three-dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.

Statement 1: if three points $P,QandR$ have position vectors $a,b,andc$ , respectively, and $2a+3b−5c=0,$ then the points $P,Q,andR$ must be collinear. Statement 2: If for three points $A,B,andC,AB=λAC,$ then points $A,B,andC$ must be collinear.

Let $a=i−k,b=xi+j +(1−x)k$ and $c=yi+xj +(1+x−y)k$ . Then $[abc]$ depends on (A) only $x$ (B) only $y$ (C) Neither $x$ nor $y$ (D) both $x$ and $y$

$A,B,CandD$ have position vectors $a,b,candd,$ respectively, such that $a−b=2(d−c)˙$ Then a. $ABandCD$ bisect each other b. $BDandAC$ bisect each other c. $ABandCD$ trisect each other d. $BDandAC$ trisect each other

Show that $∣a∣b+∣∣ b∣∣ a$ is a perpendicular to $∣a∣b−∣∣ b∣∣ a,$ for any two non-zero vectors $aandb˙$

The midpoint of two opposite sides of a quadrilateral and the midpoint of the diagonals are the vertices of a parallelogram. Prove that using vectors.

Statement 1: If $uandv$ are unit vectors inclined at an angle $αandx$ is a unit vector bisecting the angle between them, then $x=(u+v)/(2sin(α/2)˙$ Statement 2: If $DeltaABC$ is an isosceles triangle with $AB=AC=1,$ then the vector representing the bisector of angel $A$ is given by $AD=(AB+AC)/2.$