Class 12

Math

Algebra

Vector Algebra

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

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If $a=2i+3j −k,b=−i+2j −4kandc=i+j +k,$ then find thevalue of $(a×b)a×c˙ ˙$

i. Prove that the points $a−2b+3c,2a+3b−4cand−7b+10c$ are are collinear, where $a,b,c$ are non-coplanar. ii. Prove that the points $A(1,2,3),B(3,4,7),andC(−3,−2,−5)$ are collinear. find the ratio in which point C divides AB.

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 $λ˙$

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

The number of distinct real values of $λ$ , for which the vectors $λ_{2}i^+j^ +k,i^−λ_{2}j^ +k^andi^+j^ −λ_{2}k^$ are coplanar is a. zero b. one c. two d. three

If $A,B,C,D$ are four distinct point in space such that $AB$ is not perpendicular to $CD$ and satisfies $ABC˙D=k(∣∣ AD∣∣ _{2}+∣∣ BC∣∣ _{2}−∣∣ AC∣∣ _{2}−∣∣ BD∣∣ _{2}),$ then find the value of $k˙$

Let us define the length of a vector $ai^+bj^ +ck^as∣a∣+∣b∣+∣c∣˙$ This definition coincides with the usual definition of length of a vector $ai^+bj^ +ck^$ is and only if a. $a=b=c=0$ b. any two of $a,b,andc$ are zero c. any one of $a,b,andc$ is zero d. $a+b+c=0$

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ are proportional to $l_{1}+l_{2},m_{1}+m_{2},n_{1}+n_{2}˙$ Statement 2: The angle between the two intersection lines having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ is given by $cosθ=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}˙$