class 12

Math

Algebra

Vector Algebra

Let $PR=3i^+j^ −2k^andSQ=i^−3j^ −4k^$determine diagonals of a parallelogram $PQRS,andPT=i^+2j^ +3k^$be another vector. Then the volume of the parallelepiped determine by the vectors $PT$, $PQ$and $PS$is$5$b. $20$c. $10$d. $30$

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Find the scalar and vector components of the vector with initial point (2, 1) and terminal point $(5,7)$.

Find the unit vector in the direction of the sum of the vectors, $→a=2i^+2j^ −5k^$and $→b=2i^+j^ +3k^$.

Find the unit vector in the direction of vector $PQ $, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Given that $ab˙=0$and $a×b=0$. What can you conclude about the vectors $a$and $b$.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $2(a+b)$and $(a−3b)$externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ

Answer the following as true or false.(i) $a$and $−a$are collinear.(ii) Two collinear vectors are always equal in magnitude.(iii) Two vectors having same magnitude are collinear.(iv) Two collinear vectors having the same magnitude

Show that $∣a∣b+∣∣ b∣∣ a$is perpendicular to $∣a∣b−∣∣ b∣∣ a$, for any two nonzero vectors $a$ and $b$.