Class 12

Math

Algebra

Vector Algebra

Show that each of the given three vectors is a unit vector: $71 (2i^+3j^ +6k^),71 (3i^−6j^ +2k^),71 (6i^+2j^ −3k^)$Also, show that they are mutually perpendicular to each other.

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Let $a,b,andcanda_{_{′}},b_{_{′}},c_{′}$ are reciprocal system of vectors, then prove that $a_{_{′}}×b_{_{′}}+b_{_{′}}×c_{_{′}}+c_{_{′}}×a_{_{′}}=[abc]a+b+c $ .

Statement 1: If $∣∣ a+b∣∣ =∣∣ a−b∣∣ $, then $a$ and $b$ are perpendicular to each other. Statement 2: If the diagonal of a parallelogram are equal magnitude, then the parallelogram is a rectangle.

If $xandy $ are two non-collinear vectors and $ABC$ isa triangle with side lengths $a,b,andc$ satisfying $(20a−15b)x+(15b−12c)y +(12c−20a)(× xy )=0,$ then triangle $ABC$ is a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. an isosceles triangle

The vectors $2i+3j^ ,5i^+6j^ $ and 8$i^+λj^ $ have initial points at (1, 1). Find the value of $λ$ so that the vectors terminate on one straight line.

The position vectors of the vertices $A,B,andC$ of a triangle are $i^+j^ ,j^ +k^andi^+k^$ , respectively. Find the unite vector $r^$ lying in the plane of $ABC$ and perpendicular to $IA,whereI$ is the incentre of the triangle.

Prove that point $i^$ +2$j^ $ - 3$k^$ ,2$i^$ - $j^ $ + $k^$ and 2$i^$ + 5$j^ $ - $k^$ from a triangle in space.

ABC is a triangle and P any point on BC. if $PQ$ is the sum of $AP$ + $PB$ +$PC$ , show that ABPQ is a parallelogram and Q , therefore , is a fixed point.

Two forces $AB$ and $AD$ are acting at vertex A of a quadrilateral ABCD and two forces $CB$ and $CD$ at C prove that their resultant is given by 4$EF$ , where E and F are the midpoints of AC and BD, respectively.