Class 12

Math

Algebra

Vector Algebra

Show that the vectors $2i^−j^ +k^,i^−3j^ −5k^$and $3i^−4j^ −4k^$form the vertices of a right angled triangle.

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Vectors $a=i^+2j^ +3k^,b=2i^−j^ +k^$ and $c=3i^+j^ +4k^,$ are so placed that the end point of one vector is the starting point of the next vector. Then the vector are (A) not coplanar (B) coplanar but cannot form a triangle (C) coplanar and form a triangle (D) coplanar and can form a right angled triangle

Vectors $a=−4i^+3k^;b=14i^+2j^ −5k^$ are laid off from one point. Vector $d^$ , which is being laid of from the same point dividing the angle between vectors $aandb$ in equal halves and having the magnitude $6 ,$ is a. $i^+j^ +2k^$ b. $i^−j^ +2k^$ c. $i^+j^ −2k^$ d. $2i^−j^ −2k^$

Statement 1: Let $a,b,candd$ be the position vectors of four points $A,B,CandD$ and $3a−2b+5c−6d=0.$ Then points $A,B,C,andD$ are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector $(PQ,PRandPS)$ are coplanar. Then $PQ=λPR+μPS,whereλandμ$ are scalars.

The vector $a$ has the components $2p$ and 1 w.r.t. a rectangular Cartesian system. This system is rotated through a certain angel about the origin in the counterclockwise sense. If, with respect to a new system, $a$ has components $(p+1)and1$ , then $p$ is equal to a. $−4$ b. $−1/3$ c. $1$ d. $2$

Find the unit vector in the direction of the vector $a=i^+j^ +$ 2$k^$ .

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

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

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