Class 12

Math

Algebra

Vector Algebra

If $D,EandF$ are three points on the sides $BC,CAandAB,$ respectively, of a triangle $ABC$ such that the $CDBD ,AECE ,BFAF =−1$

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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.

The median AD of the triangle ABC is bisected at E and BE meets AC at F. Find AF:FC.

Let $r_{1},r_{2},r_{3},,r_{n}$ be the position vectors of points $P_{1},P_{2},P_{3},P_{n}$ relative to the origin $O˙$ If the vector equation $a_{1}r_{1}+a_{2}r_{2}++a_{n}r_{n}=0$ hold, then a similar equation will also hold w.r.t. to any other origin provided a. $a_{1}+a_{2}++˙ a_{n}=n$ b. $a_{1}+a_{2}++˙ a_{n}=1$ c. $a_{1}+a_{2}++˙ a_{n}=0$ d. $a_{1}=a_{2}=a_{3}+˙ a_{n}=0$

If $a,b,candd$ are four vectors in three-dimensional space with the same initial point and such that $3a−2b+c−2d=0$ , show that terminals $A,B,CandD$ of these vectors are coplanar. Find the point at which $ACandBD$ meet. Find the ratio in which $P$ divides $ACandBD˙$

Show that $∣a∣b+∣∣ b∣∣ a$ is a perpendicular to $∣a∣b−∣∣ b∣∣ a,$ for any two non-zero vectors $aandb˙$

Prove that $[a+bb+cc+a]=2[abc]˙$

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 the vectors $aandb$ are linearly idependent satisfying $(3 tanθ+1)a+(3 secθ−2)b=0,$ then the most general values of $θ$ are a. $nπ−6π ,n∈Z$ b. $2nπ±611π ,n∈Z$ c. $nπ±6π ,n∈Z$ d. $2nπ+611π ,n∈Z$