Class 12

Math

Algebra

Vector Algebra

Prove that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

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The sides of a parallelogram are $2i^+4j^ −5k^$ and $i^+2j^ +3k^$ . The unit vector parallel to one of the diagonals is a. $71 (3i^+6j^ −2k^)$ b. $71 (3i^−6j^ −2k^)$ c. $69 1 (i^+6j^ +8k^)$ d. $69 1 (−i^−2j^ +8k^)$

$a,b,c$ are three coplanar unit vectors such that $a+b+c=0.$ If three vectors $p ,q ,andr$ are parallel to $a,b,andc,$ respectively, and have integral but different magnitudes, then among the following options, $∣p +q +r∣$ can take a value equal to a. $1$ b. $0$ c. $3 $ d. $2$

If $a,bandc$ are three non-zero non-coplanar vectors, then find the linear relation between the following four vectors: $a−2b+3c,2a−3b+4c,3a−4b+5c,7a−11b+15⋅$

If $∣∣ a+b∣∣ <∣∣ a−b∣∣ ,$ then the angle between $aandb$ can lie in the interval a. $(π/2,π/2)$ b. $(0,π)$ c. $(π/2,3π/2)$ d. $(0,2π)$

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

Find the vector of magnitude 3, bisecting the angle between the vectors $a=2i^+j^ −k^$ and $b=i^−2j^ +k^˙$

Check whether the given three vectors are coplanar or non-coplanar. $−2i^−2j^ +4k^,−2i^+4j^ ,4i^−2j^ −2k^$

If $a,b$ are two non-collinear vectors, prove that the points with position vectors $a+b,a−b$ and $a+λb$ are collinear for all real values of $λ˙$