Class 12

Math

Algebra

Vector Algebra

If either $a=0$or $b=0$, then $a⋅b=0$But the converse need not be true. Justify your answer with an example.

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Four non –zero vectors will always be a. linearly dependent b. linearly independent c. either a or 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

If the vectors $A,B,C$ of a triangle $ABC$ are $(1,2,3),(−1,0,0),(0,1,2),$ respectively then find $∠ABC˙$

Let $r_{1},r_{2},r_{3},,r_{n}$ be the position vectors of points $P_{1},P_{2},P_{3},P_{n}$ relative to the origin $O˙$ If the vector equation $a_{1}r_{1}+a_{2}r_{2}++a_{n}r_{n}=0$ hold, then a similar equation will also hold w.r.t. to any other origin provided a. $a_{1}+a_{2}++˙ a_{n}=n$ b. $a_{1}+a_{2}++˙ a_{n}=1$ c. $a_{1}+a_{2}++˙ a_{n}=0$ d. $a_{1}=a_{2}=a_{3}+˙ a_{n}=0$

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

A vector has components $p$ and 1 with respect to a rectangular Cartesian system. The axes are rotted through an angel $α$about the origin the anticlockwise sense. Statement 1: IF the vector has component $p+2$and 1 with respect to the new system, then $p=−1.$ Statement 2: Magnitude of the original vector and new vector remains the same.

Let $a,bandc$ be three units vectors such that $2a+4b+5c=0.$ Then which of the following statement is true? a. $a$ is parallel to $b$ b. $a$ is perpendicular to $b$ c. $a$ is neither parallel nor perpendicular to $b$ d. none of these

Statement 1: In $DeltaABC,AB+BC+CA=0$ Statement 2: If $OA=a,OB=b,thenAB=a+b$