Class 12

Math

Algebra

Vector Algebra

Show that the points A(1, 2, 7), B(2, 6, 3) and $C(3,10,1)$are collinear.

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Statement 1: If $uandv$ are unit vectors inclined at an angle $αandx$ is a unit vector bisecting the angle between them, then $x=(u+v)/(2sin(α/2)˙$ Statement 2: If $DeltaABC$ is an isosceles triangle with $AB=AC=1,$ then the vector representing the bisector of angel $A$ is given by $AD=(AB+AC)/2.$

Statement 1: $a=3i+pj +3k$ and $b=2i+3j +qk$ are parallel vectors if $p=9/2andq=2.$ Statement 2: if $a=a_{1}i+a_{2}j +a_{3}kandb=b_{1}i+b_{2}j +b_{3}k$ are parallel, then $b_{1}a_{1} =b_{2}a_{2} =b_{3}a_{3} ˙$

In a trapezium, vector $BC=αAD˙$ We will then find that $p =AC+BD$ is collinear with$AD˙$ If $p =μAD,$ then which of the following is true? a. $μ=α+2$ b. $μ+α=2$ c. $α=μ+1$ d. $μ=α+1$

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 $aandb$ are two vectors of magnitude 1 inclined at $120_{0}$ , then find the angle between $bandb−a˙$

Let $A(t)=f_{1}(t)i^+f_{2}(t)j^ andB(t)=g(t)i^+g_{2}(t)j^ ,t∈[0,1],f_{1},f_{2},g_{1}g_{2}$ are continuous functions. If $A(t)andB(t)$ are non-zero vectors for all $tandA(0)=2i^+3j^ ,A(1)=6i^+2j^ ,B(0)=3i^+2i^andB(1)=2j^ +6j^ $ Then,show that $A(t)andB(t)$ are parallel for some $t$.

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

Lett $α,βandγ$ be distinct real numbers. The points whose position vector's are $αi^+βj^ +γk^;βi^+γj^ +αk^andγi^+αj^ +βk^$