Class 12

Math

Algebra

Vector Algebra

Check whether the three vectors $2i^+2j^ +3k^,−3i^+3j^ +2k^and3i^+4k^$ from a triangle or not

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A vector has component $A_{1},A_{2}andA_{3}$ in a right -handed rectangular Cartesian coordinate system $OXYZ˙$ The coordinate system is rotated about the x-axis through an angel $π/2$ . Find the component of $A$ in the new coordinate system in terms of $A_{1},A_{2},andA_{3}˙$

A boat moves in still water with a velocity which is $k$ times less than the river flow velocity. Find the angle to the stream direction at which the boat should be rowed to minimize drifting.

ABCDE is a pentagon .prove that the resultant of force $AB,$ $AE$ ,$BC$ ,$DC$ ,$ED$ and $AC$ ,is 3$AC$ .

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

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of \displaystyle\vec{{{O}{A}}},\vec{{{O}{B}}},\vec{{{O}{C}}}{\quad\text{and}\quad}\vec{{{O}{D}}} = 4 $OE$ , where O is any point.

In a triangle $ABC,DandE$ are points on $BCandAC,$ respectivley, such that $BD=2DCandAE=3EC˙$ Let $P$ be the point of intersection of $ADandBE˙$ Find $BP/PE$ using the vector method.

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. $(1,1,2)$

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