Class 12

Math

Algebra

Vector Algebra

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and$Q(4,1,2)$.

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If $a,b,andc$ are mutually perpendicular vectors and $a=α(a×b)+β(b×c)+γ(c×a)and[abc]=1,$ then find the value of $α+β+γ˙$

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of \displaystyle\vec{{{O}{A}}},\vec{{{O}{B}}},\vec{{{O}{C}}}{\quad\text{and}\quad}\vec{{{O}{D}}} = 4 $OE$ , where O is any point.

Find the direction cosines of the vector joining the points $A(1,2,−3)a∩B(−1−2,1)$ directed from $A→B˙$

Let $u^=i^+j^ ,v^=i^−j^ andw^=i^+2j^ +3k^˙$ If $n^$ is a unit vector such that $u^n^˙=0andv^n^˙=0,$ then find the value of $∣∣ w^n^˙∣∣ ˙$

Sow that $x_{1}i^+y_{1}j^ +z_{1}k^,x_{2}i^+y_{2}j^ +z_{2}k^,andx_{3}i^+y_{3}j^ +z_{3}k^,$ are non-coplanar if $∣x_{1}∣>∣y_{1}∣+∣z_{1}∣,∣y_{2}∣>∣x_{2}∣+∣z_{2}∣and∣z_{3}∣>∣x_{3}∣+∣y_{3}∣$ .

Show, by vector methods, that the angularbisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

$a,b,c$ 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. $2$

A man travelling towards east at 8km/h finds that the wind seems to blow directly from the north On doubling the speed, he finds that it appears to come from the north-east. Find the velocity of the wind.