Class 12

Math

Algebra

Vector Algebra

If either $a=0$and $b=0$then $a×b=0$. Is Is the converse true? Justify your answer with an example.

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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 $b_{1}a_{1} =b_{2}a_{2} =b_{3}a_{3} ˙$

Find the values of $λ$ such that $x,y,z=(0,0,0)and(i^+j^ +3k^)x+(3i^−3j^ +k^)y+(−4i^+5j^ )z=λ(xi^+yj^ +zk^)$, where $i^,j^ ,k^$ are unit vector along coordinate axes.

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. $μ=α+1$

If $aandb$ are two unit vectors and $θ$ is the angle between them, then the unit vector along the angular bisector of $a$ and $b$ will be given by a. $cos(θ/2)a−b $ b. $2cos(θ/2)a+b $ c. $2cos(θ/2)a−b $ d. none of these

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 $cosθ=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}˙$

Find a vector in the direction of the vector 5$i^$ - $j^ $ + 2$k^$ which has magnitude 8 units.

Show that the vectors $2a−b+3c,a+b−2canda+b−3c$ are non-coplanar vectors (where $a,b,c$ are non-coplanar vectors)

Given that $ab˙=ac˙,a×b=a×canda$ is not a zero vector. Show that $b=⋅$