Class 12

Math

Algebra

Vector Algebra

If the resultant of two forces is equal in magnitude to one of the components and perpendicular to it direction, find the other components using the vector method.

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The position vectors of the vertices $A,BandC$ of a triangle are three unit vectors $a,b,andc,$ respectively. A vector $d$ is such that $da˙=db˙=dc˙andd=λ(b+c)˙$ Then triangle $ABC$ is a. acute angled b. obtuse angled c. right angled d. none of these

Let $a,bandc$ be unit vectors such that $a+b−c=0.$ If the area of triangle formed by vectors $aandbisA,$ then what is the value of $4A_{2}?$

if $Ao$ + $OB$ = $BO$ + $OC$ ,than prove that B is the midpoint of AC.

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

Three coinitial vectors of magnitudes a, 2a and 3a meet at a point and their directions are along the diagonals if three adjacent faces if a cube. Determined their resultant R. Also prove that the sum of the three vectors determinate by the diagonals of three adjacent faces of a cube passing through the same corner, the vectors being directed from the corner, is twice the vector determined by the diagonal of the cube.

If $a,bandc$ are non-coplanar vectors, prove that the four points $2a+3b−c,a−2b+3c,3a+$ 4$b−2canda−6b+6c$ are coplanar.

OABCDE is a regular hexagon of side 2 units in the XY-plane in the first quadrant. O being the origin and OA taken along the x-axis. A point P is taken on a line parallel to the z-axis through the centre of the hexagon at a distance of 3 unit from O in the positive Z direction. Then find vector AP.

In triangle $ABC,∠A=30_{0},H$ is the orthocenter and $D$ is the midpoint of $BC$. Segment $HD$ is produced to $T$ such that $HD=DT$ The length $AT$ is equal to (a). $2BC$ (b). $3BC$ (c). $24 BC$ (d). none of these