Class 12

Math

Algebra

Vector Algebra

Let the vectors $a$and $b$be such that $∣a∣=3$and $∣∣ b∣∣ =32 $, then $a×b$is a unit vector, if the angle between $a$and $b$(A) $π/6$ (B) $π/4$ (C) $π/3$ (D) $π/2$

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OABCDE is a regular hexagon of side 2 units in the XY-plane in the first quadrant. O being the origin and OA taken along the x-axis. A point P is taken on a line parallel to the z-axis through the centre of the hexagon at a distance of 3 unit from O in the positive Z direction. Then find vector AP.

If non-zero vectors $aandb$ are equally inclined to coplanar vector $c,thenc$ can be a. $∣a∣+2∣∣ b∣∣ ∣a∣ a+∣a∣+∣∣ b∣∣ ∣∣ b∣∣ b$ b. $∣a∣+∣∣ b∣∣ ∣∣ b∣∣ a+∣a∣+∣∣ b∣∣ ∣a∣ b$ c. $∣a∣+2∣∣ b∣∣ ∣a∣ a+∣a∣+2∣∣ b∣∣ ∣∣ b∣∣ b$ d. $2∣a∣+∣∣ b∣∣ ∣∣ b∣∣ a+2∣a∣+∣∣ b∣∣ ∣a∣ b$

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

Statement 1:Let $A(a),B(b)andC(c)$ be three points such that $a=2i^+k^,b=3i^−j^ +3k^andc=−i^+7j^ −5k^˙$ Then $OABC$ is a tetrahedron. Statement 2: Let $A(a),B(b)andC(c)$ be three points such that vectors $a,bandc$ are non-coplanar. Then $OABC$ is a tetrahedron where $O$ is the origin.

Prove that $[a+bb+cc+a]=2[abc]˙$

Statement 1: If $∣∣ a+b∣∣ =∣∣ a−b∣∣ $, then $a$ and $b$ are perpendicular to each other. Statement 2: If the diagonal of a parallelogram are equal magnitude, then the parallelogram is a rectangle.

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

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.