class 12

Math

Algebra

Vector Algebra

The resultant of two forces P N and 3 N is a force of 7 N. If the direction of 3 N force were reversed, the resultant would be $19 $N. The value of P is

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Prove that vectors $u=(al+a_{1}l_{1})i^+(am+a_{1}m_{1})j^ +(an+a_{1}n_{1})k^$ $v=(bl+b_{1}l_{1})i^+(bm+b_{1}m_{1})j^ +(bn+b_{1}n_{1})k^$ $w=(bl+b_{1}l_{1})i^+(bm+b_{1}m_{1})j^ +(bn+b_{1}n_{1})k^$ are coplanar.

If $a,bandc$ are non-cop0lanar vector, then that prove $∣∣ (ad˙)(b×c)+(bd˙)(c×a)+(cd˙)(a×b)∣∣ $ is independent of $d,wheree$ is a unit vector.

Given three points are $A(−3,−2,0),B(3,−3,1)andC(5,0,2)˙$ Then find a vector having the same direction as that of $AB$ and magnitude equal to $∣∣ AC∣∣ ˙$

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

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

A pyramid with vertex at point $P$ has a regular hexagonal base $ABCDEF$ , Positive vector of points A and B are $i^andi^+2j^ $ The centre of base has the position vector $i^+j^ +3 k^˙$ Altitude drawn from $P$ on the base meets the diagonal $AD$ at point $G˙$ find the all possible position vectors of $G˙$ It is given that the volume of the pyramid is $63 $ cubic units and $AP$ is 5 units.

If vectors $AB=−3i^+4k^andAC=5i^−2j^ +4k^$ are the sides of a$DeltaABC,$ then the length of the median through $Ais$ a. $14 $ b. $18 $ c. $29 $ d. $5 $

Let $D,EandF$ be the middle points of the sides $BC,CAandAB,$ respectively of a triangle $ABC˙$ Then prove that $AD+BE+CF=0$ .