CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 5 - MCQExams.com

A man starts from $$O$$ moves $$500m$$ turns by $$60 ^\circ$$ and moves $$500m$$ again turns by $$60 ^\circ$$ and moves $$500m$$ and so on. Find the displacement after $$(i)$$ 5th turn , $$(ii)$$ 3rd turn.
  • -$$500m, 1000m$$
  • $$500m, 500 \sqrt {3} m$$
  • $$1000m, 500 \sqrt {3} m$$
  • none of these
The projection of the vector $$\hat i - 2\hat j + \hat k$$ on the vector $$4\hat i - 4\hat j + 7\hat k$$ is
  • $$\displaystyle \dfrac{5\sqrt{6}}{10}$$
  • $$\displaystyle \dfrac{19}{9}$$
  • $$\displaystyle \dfrac{9}{19}$$
  • $$\displaystyle \dfrac{\sqrt{6}}{19}$$
If the vector product of a constant vector $$\displaystyle \vec{OA}$$ with a variable vector $$\displaystyle \vec{OB}$$ in a fixed plane $$OAB$$ be a constant vector, then locus of $$B$$ is :
  • A straight line perpendicular to $$\displaystyle \vec{OA}$$
  • A circle with centre O radius equal to $$\displaystyle \left | \vec{OA} \right |$$
  • A straight line parallel to $$\displaystyle \vec{OA}$$
  • None of these
There are $$N$$ co-planar vectors each of magnitude $$V$$. Each vector is inclined to the preceding vector at angle $$2 \pi/N$$. What is the magnitude of their resultant?
  • zero
  • $$V/N$$
  • $$V$$
  • $$NV$$
$$P, Q$$ have position vectors $$\vec {a}$$ & $$\vec {b}$$ relative to the origin '$$O$$' and $$ X, Y$$  divide $$\vec{PQ}$$ intermally and extemally respectively in the ratio $$2:1$$, then vector $$\vec{XY}= $$
  • $$\displaystyle \:\dfrac{3}{2}(\vec b - \vec a)$$
  • $$\displaystyle \:\dfrac{4}{3}(\vec c - \vec a)$$
  • $$\displaystyle \:\dfrac{5}{6}\left ( \vec{b}- \vec{a}\right )$$
  • $$\displaystyle \:\dfrac{4}{3}(\vec b - \vec a)$$
If . and $$\times $$ represent dot product and cross product respectively then which of the following is meaningless?
  • $$\left ( \vec{a}\times \vec{b} \right ).\left ( \vec{c}\times \vec{d} \right )$$
  • $$\left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )$$
  • $$\left ( \vec{a}.\vec{b} \right ) .\left ( \vec{c}\times \vec{d} \right )$$
  • $$\left ( \vec{a}.\vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )$$
If $$\vec{a}.\vec{b}=0$$ and $$\vec{a}\times \vec{b}=0$$ then
  • $$\vec{a}\parallel \vec{b}$$
  • $$\vec{a}\perp \vec{b}$$
  • $$\vec{a}=\vec{0}$$ or $$\vec{b}=\vec{0}$$
  • None of these
If $$G$$ and $$G'$$ be the centroids of the triangles $$ABC$$ and $$A'B'C'$$ respectively, then $$\overrightarrow { AA' } +\overrightarrow { BB' } +\overrightarrow { CC' } =$$
  • $$\displaystyle \dfrac { 2 }{ 3 } \overrightarrow { GG' } $$
  • $$\overrightarrow { GG' } $$
  • $$2\overrightarrow { GG' } $$
  • $$3\overrightarrow { GG' } $$
Five coplanar forces of equal magnitudes $$10 N$$ each, act at a point such that the angle between any two consecutive forces is same. The magnitude of their resultant is :
  • $$0$$
  • $$10 N$$
  • $$20 N$$
  • $$10 \sqrt {2} N$$
$$P$$ is a point on the line through the point $$A$$ whose position vector is $$\overrightarrow{a}$$ and the line is parallel to the vector $$\overrightarrow{b}$$. If $$PA=6$$, the position vector of $$P$$ is
  • $$\overrightarrow{a}+6\overrightarrow{b}$$
  • $$\displaystyle \overrightarrow{a}+\dfrac{6}{\left |\overrightarrow{b} \right |}\overrightarrow{b}$$
  • $$\overrightarrow{a}-6\overrightarrow{b}$$
  • $$\displaystyle \overrightarrow{b}+\dfrac{6}{\left |\overrightarrow{a} \right |}\overrightarrow{a}$$
A straight line $$r=a+\lambda b$$ meets the plane $$r.n=0$$ in $$P$$. The position vector of $$P$$ is
  • $$\displaystyle a+\frac { a.n }{ b.n } b$$
  • $$\displaystyle a+\frac { b.n }{ a.n } b$$
  • $$\displaystyle a-\frac { a.n }{ b.n } b$$
  • None of these
Let $$\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}$$. If the point $$P$$ on the line segment $$BC$$ is equidistant from $$AB$$ and $$AC$$ then $$\overrightarrow{AP}$$ is
  • $$2\hat{i}-\hat{k}$$
  • $$\hat{i}-2\hat{k}$$
  • $$2\hat{i}+\hat{k}$$
  • None of these
If the vectors $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ and $$\vec{d}$$ are coplanar, then $$\left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )$$ is equal to
  • $$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$
  • $$\vec{0}$$
  • $$\vec{a}+\vec{b}=\vec{c}+\vec{d}$$
  • None of these
If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$, $$\vec{d}$$ are any four vectors then $$\left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )$$ is a vector
  • Perpendicular to $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$, $$\vec{d}$$
  • Along the line of intersection of two planes, one containing $$\vec{a}$$, $$\vec{b}$$ and the other containing $$\vec{c}$$, $$\vec{d}$$
  • Equally inclined to both $$\vec{a}\times \vec{b}$$ and $$\vec{c}\times \vec{d}$$
  • None of these
The position vectors of two vertices and the centroid of a triangle are $$\vec{i}+\vec{j}$$, $$2\vec{i}-\vec{j}+\vec{k}$$ and $$\vec{k}$$ respectively, then the position vector of the third vertex of the triangle is 
  • $$-3\vec{i}+2\vec{k}$$
  • $$3\vec{i}-2\vec{k}$$
  • $$\vec{i}+\dfrac{2}{3}\vec{k}$$
  • None of these
$$\displaystyle R(\bar{r})$$ is any point on a semi-circle, $$\displaystyle P(\bar{p})$$ and $$\displaystyle Q(\bar{q})$$ are the position vector of the end point of the diameter of that semi-circle, then $$\displaystyle \overline{PR}\cdot \overline{QR}$$ is equal to
  • $$1$$
  • $$0$$
  • $$3$$
  • Not defined
In a triangle OAB, E is the mid-point of OB and D is a point on AB such that AD: DB = 2 :If OD and AE intersect at P, determine the ratio OP : PD using vector methods.
  • $$OP:PD=2:3$$
  • $$OP:PD=3:2$$
  • $$OP:PD=1:3$$
  • $$OP:PD=3:1$$
In a triangle ABC, D divides BC in the ratio 3 : 2 and E divides CA in the ratio 1 :The lines AD and BE meet at H and CH meets AB in F. Find the ratio in which F divides AB.
  • $$\displaystyle AF:FB=2:1$$
  • $$\displaystyle AF:FB=1:2$$
  • $$\displaystyle AF:FB=2:3$$
  • $$\displaystyle AF:FB=3:2$$
$$\displaystyle \bar{a}= x\hat{i}+y\hat{j}+z\hat{k},\bar{b}= \hat{j}$$ then the vector $$\displaystyle \bar{c}$$ for which $$\displaystyle \bar{a},\bar{b},\bar{c}$$ form a right hand triad
  • $$\displaystyle x(\hat{i}-\hat{k})$$
  • $$\displaystyle \bar{0}$$
  • $$\displaystyle -z\hat{i}+x\hat{k}$$
  • $$\displaystyle y\hat{j}$$
Let $$\displaystyle \bar{p}$$ and $$\displaystyle \bar{q}$$ be two distinct points. Let $$R$$ and $$S$$ be the points dividing $$PQ$$ internally and externally in the ratio $$2:3$$. If $$\displaystyle \overline{OR} \perp  \overline{OS},$$ then
  • $$\displaystyle 9p^{2}= 4q^{2}$$
  • $$\displaystyle 4p^{2}= 9q^{2}$$
  • $$\displaystyle 9p= 4q $$
  • $$\displaystyle 4p= 9q $$
The difference of the squares on the diagonals is four times the rectangle contained by either of these sides and the projection of the other upon it.
  • $$\displaystyle = 4$$ ( rectangle contained by AB and projection of AC on AB).
  • $$\displaystyle = 1$$ ( rectangle contained by AB and projection of AC on AB).
  • $$\displaystyle = 8$$ ( rectangle contained by AB and projection of AC on AB).
  • $$\displaystyle = 2$$ ( rectangle contained by AB and projection of AC on AB).
State true or false:
The four diagonals of a parallelopiped and the joins of the mid-points of opposite edges are concurrent at a common point of bisection.
  • True
  • False
$$ABC$$ is a $$\displaystyle \Delta $$ and $$G$$ is its centroid. If $$\displaystyle \overline{AB}= \bar{b}$$ and $$\displaystyle \overline{AC}= \bar{c}$$, then $$\displaystyle \overline{AG}$$ is equal to
  • $$\displaystyle \dfrac{2}{3}\left ( \bar{b}+\bar{c} \right )$$
  • $$\displaystyle \bar{b}+\bar{c} $$
  • $$\displaystyle \dfrac{1}{3}\left ( \bar{b}+\bar{c} \right )$$
  • $$\displaystyle \dfrac{3}{2}\left ( \bar{b}+\bar{c} \right )$$
In a parallelogram $$ABCD,\left| AB \right| =a,\left| AD \right| =b$$ and $$\left| AC \right| =c.$$ Then, $$DB.AB$$ has the value
  • $$\displaystyle\frac { 3{ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle\frac { { a }^{ 2 }+3{ b }^{ 2 }-{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle\frac { { a }^{ 2 }-{ b }^{ 2 }+3{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle\frac { { a }^{ 2 }+3{ b }^{ 2 }+{ c }^{ 2 } }{ 2 } $$
Let $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ be vectors with magnitudes $$3, 4$$ and $$5$$ respectively and $$\bar{a}+\bar{b}+\bar{c}=0$$, then the value of $$\bar{a}.\bar{b}+\bar{b}.\bar{c}+\bar{c}.\bar{a}$$ is
  • $$48$$
  • $$-26$$
  • $$25$$
  • $$-25$$
The sides of a $$\triangle$$ are in A.P, then the line joining the centroid to the incenter is parallel to
  • The largest side
  • The middle side
  • The smallest side
  • None of these
If $$\bar{\alpha }$$, $$\bar{\beta }$$ and $$\bar{\gamma }$$ be vertices of a $$\triangle $$ whose circumcenter is at the origin, then orthocenter is given by
  • $$\displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{4}$$
  • $$\displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{2}$$
  • $$\bar{\alpha }+\bar{\beta }+\bar{\gamma }$$
  • $$\displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{3}$$
If $$S$$ is the circumcenter, $$O$$ is the orthocenter of $$\triangle ABC$$, then $$\displaystyle \vec{SA}+\vec{SB}+\vec{SC}= $$
  • $$2\vec{OS}$$
  • $$2\vec{SO}$$
  • $$\vec{OS}$$
  • $$\vec{SO}$$
For non-zero vectors $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$, $$\left | \left ( \bar{a}\times \bar{b} \right ).\bar{c} \right |=\left | \bar{a} \right |\left | \bar{b} \right |\left | \bar{c} \right |$$ iff
  • $$\bar{a}.\bar{c}=0$$, $$\bar{a}.\bar{b}=0$$
  • $$\bar{a}.\bar{b}=0$$, $$\bar{b}.\bar{c}=0$$
  • $$\bar{c}.\bar{a}=0$$, $$\bar{b}.\bar{c}=0$$
  • $$\bar{a}.\bar{b}=\bar{b}.\bar{c}=\bar{c}.\bar{a}=0$$
Two planes are perpendicular to each other,one of them contains vector $$\bar{a}$$ and $$\bar{b}$$, other contains $$\bar{c}$$ and $$\bar{d}$$ then $$\left ( \bar{a}\times \bar{b} \right )\cdot \left (\bar{c}\times \bar{d}  \right )=$$
  • $$1$$
  • $$0$$
  • $$\left [ \bar{a}\bar{b}\bar{c} \right ]$$
  • $$\left [ \bar{b}\bar{c}\bar{d} \right ] $$
Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $$\bar{OA}+\bar{OB}+\bar{OC}+\bar{OD}=$$
  • $$2\bar{OP} $$
  • $$3\bar{OP} $$
  • $$\bar{OP} $$
  • $$4\bar{OP}$$
If a. $$\displaystyle b\neq 0,$$ find the vector r which satisfies the equations $$\displaystyle \left ( r-c \right )\times b= 0, r.a= 0$$
  • $$\displaystyle r= \frac{\left ( a.b \right )c+\left ( a.c \right )b}{a.b}$$
  • $$\displaystyle r= \frac{\left ( a.b \right )c-\left ( a.c \right )b}{a.b}$$
  • $$\displaystyle r= \frac{-\left ( a.b \right )c-\left ( a.c \right )b}{a.b}$$
  • $$\displaystyle r= \frac{-\left ( a.b \right )c+\left ( a.c \right )b}{a.b}$$
If $$\alpha \bar{a}+\beta \bar{b}+\gamma \bar{c}=0$$, then $$\left ( \bar{a}\times \bar{b} \right )\times \left [ \left ( \bar{b}\times \bar{c} \right )\times \left ( \bar{c}\times \bar{a} \right ) \right ]$$ is equal to
  • $$\bar{0}$$
  • A vector $$\perp $$ plane of $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$
  • A scalar quantity
  • None of these
A unit vector perpendicular to each of the vectors $$\displaystyle 2\hat{i}+4\hat{j}-\hat{k} $$ and $$\displaystyle \hat{i}-2\hat{j}+3\hat{k} $$ forming a right handed system is
  • $$\displaystyle 7\hat{i}-10\hat{j}+8\hat{k}$$
  • $$\displaystyle \frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$$
  • $$\displaystyle -7\hat{i}+10\hat{j}+8\hat{k}$$
  • $$\displaystyle \frac{-10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$$
The position vector of the points $$A$$ and $$B$$ are respectively $$\bar{a}$$ and $$\bar{b}$$  divides $$AB$$ in the ratio $$3:1$$ and $$Q$$ iis the midpoint of $$AP$$. The position vector of $$Q$$ is
  • $$\displaystyle \dfrac{3\bar{a}+5\bar{b}}{8}$$
  • $$\displaystyle\dfrac{3\bar{a}+\bar{b}}{4}$$
  • $$\displaystyle\dfrac{\bar{a}+3\bar{b}}{4}$$
  • $$\displaystyle\dfrac{5\bar{a}+3\bar{b}}{8}$$
For three unit vectors $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$, if $$\bar{a}+\bar{b}+\bar{c}= \bar{0}$$, then the value of $$\bar{a}\cdot \left ( \bar{b}+\bar{c} \right )+\bar{b}\cdot \left ( \bar{c}+\bar{a} \right )+\bar{c}\cdot \left ( \bar{a}+\bar{b} \right )$$ is equal to
  • $$-\dfrac{3}{2}$$
  • $$0$$
  • $$-3$$
  • None of these
If $$ \displaystyle \bar{a}\times \bar{b}=\bar{b}\times \bar{c}=\bar{c}\times \bar{a} $$ then $$ \displaystyle \bar{a}+\bar{b}+\bar{c}=? $$
  • $$abc$$
  • $$-1$$
  • $$0$$
  • $$2$$
The vector $$\left ( \bar{a }\times\bar{b } \right )\times \left ( \bar{c }\times\bar{b } \right )$$ is
  • At right angle to $$\bar{b}$$
  • Is parallel to $$\bar{c}$$
  • Is parallel to $$\bar{b}$$
  • None of these
In a trapezium the vector $$\overline{BC} = \alpha \overline{AD}$$. We will then find that $$\bar{p}= \overline{AC}+\overline{BD}$$ is collinear with $$\overline{AD}$$. if $$\bar{p}= \mu \overline{AD}$$ then
  • $$\mu =\alpha +2$$
  • $$\mu +\alpha = 2$$
  • $$\alpha = \mu +1$$
  • $$\mu = \alpha+1$$
If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB$$, then
  • $$ \displaystyle \overline{PA}+\overline{PB}+\overline{PC}=0 $$
  • $$ \displaystyle \overline{PA}+\overline{PB}+2\overline{PC}=\bar{0} $$
  • $$ \displaystyle \overline{PA}+\overline{PB}=\overline{PC} $$
  • $$ \displaystyle \overline{PA}+\overline{PB}=2\overline{PC} $$
Given $$A=ai+bj+ck, \ B=di+3j+4k$$ and $$C=3i+j-2k$$. If the vectors $$A,B$$ and $$C$$ form a triangle such that $$A=B+C$$ and $$area(\Delta ABC)=5\sqrt6$$, then
  • $$ a=-8, \ b=-4, \ c=2, \ d=-11 $$
  • $$a=-8, \ b=4, \ c=-2, \ d=-11$$
  • $$a=-8,\ b=4, \ c=2, \ d=-11$$
  • None of the above
Two identical particles are located at $$\overrightarrow{x}$$ and $$ \overrightarrow{y}$$ with reference to the origin of three dimensional co-ordinate system. The position vector of centre of mass of the system is given by
  • $$\overrightarrow { x } - \overrightarrow { y }$$
  • $$\displaystyle \frac {\overrightarrow { x } + \overrightarrow { y }} {{2}}$$
  • $$(\overrightarrow { x } - \overrightarrow { y })$$
  • $$\displaystyle \frac{\overrightarrow { x } - \overrightarrow { y }} {{2}}$$
A vector $$a$$ can be written as
  • $$(a\cdot i)i+(a\cdot j)j+(a\cdot k)k$$
  • $$(a\cdot j)i+(a\cdot k)j+(a\cdot i)k$$
  • $$(a\cdot k)j+(a\cdot i)j+(a\cdot j)k$$
  • $$(a\cdot a)i+(i+j+k) k$$
If $$b \neq 0,$$ then every vector $$a$$ can be written in a unique manner as the sum of a vector $$a_{||}$$ parallel to b and a vector $$a_{\perp}$$ perpendicular to $$b$$. If $$a$$ is parallel to $$ b,$$ then $$a_{||} = a $$ and $$ a_{\perp } =0$$. If $$a$$ is perpendicular to $$b$$, then $$a_{||} = 0$$ and $$a_{\perp} = a$$. The vector $$a_{||}$$ is called the projection of $$a$$ on $$b$$ and is denoted by $$proj_{b}a$$. Since $$proj_{b} a$$ is parallel to $$b$$, it is a scalar multiple of the unit vector in the direction of $$b$$, i.e., $$proj _{b} a =\lambda u_{b}$$
The scalar $$\lambda $$ is called the component of $$a$$ in the direction of $$b$$ and is denoted by $$comp _{b} a $$. In fact, $$proj _{b} a = (a \cdot u_{b})u_{b}$$ and $$comp _{b} a = a \cdot u_{b}$$

If $$a = -2i + j + k $$ and $$ b =4i -3j + k $$, then $$proj_{b} a$$ is equal to
  • $$ 4i -3j + 2k$$
  • $$\displaystyle -\frac{ 5}{13} (4i- 3j + k)$$
  • $$\displaystyle \frac{ 5}{13} (4i -3j+ k)$$
  • $$\displaystyle -\frac{ 4}{11} (2i -j + 2k)$$
$$ABCD$$ is a quadrilateral and $$E$$ the point of intersection of the lines joining the middle points of opposite sides. If $$O$$ is any point, then the resultant of $$OA,OB,OC$$ and $$OD$$ is equal to
  • $$2OE$$
  • $$OE$$
  • $$4OE$$
  • none of these
The components of vector $$ i + j + k $$ along vector $$i+2j+3k $$ is
  • $$(3/7) (i + 2j + 3k)$$
  • $$(i + 2j + 3k)$$
  • $$(1/7) (i + 2j + 3k)$$
  • $$(4/7) (i + 2j + 3k)$$
Four forces of magnitude P, 2P, 3P and 4P act along the four sides of a square ABCD in cyclic order. Use the vector method to find the resultant force.
  • $$\displaystyle 2\sqrt{2}P$$
  • $$\displaystyle 3\sqrt{2}P$$
  • $$\displaystyle \sqrt{2}P$$
  • $$\displaystyle -2\sqrt{2}P$$
OAB is a given triangle such that $$\displaystyle \overrightarrow{OA}=\bar a$$, $$\displaystyle \overrightarrow{OB}=\bar b$$.
Also C is a point on AB such that $$\overrightarrow{AB}=2\overrightarrow{BC}$$. State which of the following statements are correct?
  • $$\displaystyle \overrightarrow{AC}=\frac{2}{3}\left ( \bar b-\bar a \right )$$,
  • $$\displaystyle \overrightarrow{AC}=\frac{2}{3}\left ( \bar a-\bar b \right )$$,
  • $$\displaystyle \overrightarrow{AC}=\frac{2}{3}\left (\bar b+\bar a \right )$$,
  • $$\displaystyle \overrightarrow{AC}=\frac{3}{2}\left ( \bar b-\bar a \right )$$,
$$ABC$$ is any triangle and $$D, E, F$$ are the middle points of its sides $$BC, CA, AB$$ respectively. Express the vectors $$\displaystyle \overrightarrow{CF}$$ and $$\overrightarrow{BE}$$ as linear combination of the vectors $$\displaystyle \overrightarrow{AB}$$ and $$\overrightarrow{AC}$$
  • $$\displaystyle \overrightarrow{BE}= \frac{1}{2}\overrightarrow{AC}-\overrightarrow{AB};\,\,\,\overrightarrow{CF}= \frac{1}{2}\overrightarrow{AB}-\overrightarrow{AC}$$
  • $$\displaystyle \overrightarrow{BE}= \frac{1}{2}\overrightarrow{AC}+\overrightarrow{AB};\,\,\,\overrightarrow{CF}= \frac{1}{2}\overrightarrow{AB}+\overrightarrow{AC}$$
  • $$\displaystyle \overrightarrow{CF}= \frac{1}{2}\overrightarrow{AC}-\overrightarrow{AB};\,\,\,\overrightarrow{BE}= \frac{1}{2}\overrightarrow{AB}-\overrightarrow{AC}$$
  • $$\displaystyle \overrightarrow{CF}= \frac{1}{2}\overrightarrow{AC}+\overrightarrow{AB};\,\,\,\overrightarrow{BE}= \frac{1}{2}\overrightarrow{AB}+\overrightarrow{AC}$$
If $$A\left( \overrightarrow { a }  \right) ,B( \overrightarrow { b } ) ,C\left( \overrightarrow { c }  \right) $$ be the vertices of a triangle whose circumcentre is the origin, then orthocenter is given by
  • $$\displaystyle \dfrac { \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }  }{ 3 } $$
  • $$\displaystyle \dfrac { \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }  }{ 2 } $$
  • $$\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } $$
  • None of these
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