Class 12

Math

Algebra

Vector Algebra

Write two different vectors having same magnitude.

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If $a=5i^−j^ −3k^$and $b=i^+3j^ −5k^$then show that the vectors $a+b$and $a−b$are perpendicular.

If $θ$ is the angle between any two vectors $a$ and $b$, then $∣∣ a.b∣∣ =∣∣ a×b∣∣ $ when $θ$ is equal to(A) 0 (B) $4π $ (C) $2π $ (D) $π$

Find the area of the parallelogram whose adjacent sides are determined by the vectors $a=i^−j^ +3k^$ and $b=2i^−7j^ +k^$.

Show that the points $A(−2i^+3j^ +5k^),B(i^+2j^ +3k^)$and $C(7i^−3k^)$are collinear.

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

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

Find the direction cosines of the vector joining the points $A(1,2,3)$and$B(1,2,1)$, directed from A to B.

Find $∣a∣$and $∣∣ b∣∣ $, if $(a+b)(a−b)˙ =8$and $∣a∣=8∣∣ b∣∣ $