CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 6 - MCQExams.com

The projection of $$\bar a=3\hat i-\hat j +5\hat k$$ on $$\bar b=2\hat i+3\hat j+\hat k$$ is
  • $$\displaystyle 8/\sqrt{(35)}$$
  • $$\displaystyle 8/\sqrt{(39)}$$
  • $$\displaystyle 8/\sqrt{(14)}$$
  • $$\displaystyle \sqrt{(14)}$$
The triangle $$ABC$$ is defined by the vertices $$A (1, -2, 2), B(1, 4, 0)$$ and $$C(-4, 1, 1)$$. Let $$M$$ be the foot of the altitude drawn from the vertices $$B$$ to side $$AC$$. Then $$\displaystyle \overrightarrow{BM}=$$
  • $$\left (-\dfrac {20}{7}, -\dfrac {30}{7}, \dfrac {10}{7}\right)$$
  • $$(-20, -30, 10)$$
  • $$(2, 3, -1)$$
  • none of these
Given two vectors $$a=2\hat{i}-3\hat{j}+6\hat{k}$$, $$b=-2\hat{i}+2\hat{j}-1\hat{k}$$ and $$\displaystyle \lambda$$ = ratio of the projection of $$a$$ on $$b$$ and the projection of $$b$$ on $$a,$$ then the value of $$\displaystyle \lambda $$ is
  • $$\dfrac{3}{7}$$
  • $$\dfrac{7}{3}$$
  • $$3$$
  • $$7$$
If c is the middle point of AB and P is any point outside AB then
  • $$\displaystyle \overrightarrow{PA}+\overrightarrow{PB}=\overrightarrow{PC}$$
  • $$\displaystyle \overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PC}$$
  • $$\displaystyle \overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=0$$
  • $$\displaystyle \overrightarrow{PA}+\overrightarrow{PB}+2\overrightarrow{PC}=0$$
Let $$a=i+2j+k, \ b=i-j+k$$ and $$c=i+j-k $$. A vector in the plane of $$ a$$ and $$b$$, whose projection on $$c$$ is $$\displaystyle \frac{1}{\sqrt{3}}$$ is
  • $$-4i+j-4k$$
  • $$3i+j-3k$$
  • $$ i+j-2k$$
  • $$4i+j-4k$$
ABCD is a parallelogram and AC, BD are its diagonals Express $$\displaystyle \overrightarrow{AC}\:and\:\overrightarrow{BD}$$ in terms of $$\displaystyle \overrightarrow{AB}\:and\:\overrightarrow{AD}$$ only
  • $$\displaystyle \overrightarrow{AC}= \overrightarrow{AB}+\overrightarrow{AD}\quad;\:\overrightarrow{BD}= -\overrightarrow{AB}+\overrightarrow{AD}$$
  • $$\displaystyle \overrightarrow{AC}= \overrightarrow{AB}-\overrightarrow{CB}\quad;\:\overrightarrow{BD}= -\overrightarrow{AB}+\overrightarrow{AD}$$
  • $$\displaystyle \overrightarrow{AC}= \overrightarrow{AB}+\overrightarrow{AD}\quad;\:\overrightarrow{BD}= \overrightarrow{BC}+\overrightarrow{CD}$$
  • all of the above
Given a cube $$ABCD{ A }_{ 1 }{ B }_{ 1 }{ C }_{ 1 }{ D }_{ 1 }$$ with lower base $$ABCD$$, upper base $${ A }_{ 1 }{ B }_{ 1 }{ C }_{ 1 }{ D }_{ 1 }$$ and the lateral edges $$A{ A }_{ 1 },B{ B }_{ 1 },C{ C }_{ 1 }$$ and $$D{ D }_{ 1 }$$; $$M$$ and $${ M }_{ 1 }$$ are the centers of the faces $$ABCD$$ and $${ A }_{ 1 }{ B }_{ 1 }{ C }_{ 1 }{ D }_{ 1 }$$ respectively. $$O$$ is apoint on line $$M{ M }_{ 1 }$$, such that
$$OA+OB+OC+OD=O{ M }_{ 1 }$$, then $$OM=\lambda O{ M }_{ 1 }$$ is $$\lambda=$$
  • $$\displaystyle \frac { 1 }{ 4 } $$
  • $$\displaystyle \frac { 1 }{ 2 } $$
  • $$\displaystyle \frac { 1 }{ 6 } $$
  • $$\displaystyle \frac { 1 }{ 8 } $$
$$P, Q, R, S$$ have position vectors $$p, q, r, s$$, such that $$\displaystyle p-q=2\left ( s-r \right )$$, then
  • $$PQ$$ and $$RS$$ bisect each other
  • $$PQ$$ and $$PR$$ bisect each other
  • $$PQ$$ and $$RS$$ trisect each other
  • $$QS$$ and $$PR$$ trisect each other
Which of the following statements are correct:
If $$M$$ is the mid-point of $$AB$$ and $$O$$ is any point then
  • $$\displaystyle \overrightarrow{OM}= \overrightarrow{OA}+\overrightarrow{MA}$$
  • $$\displaystyle \overrightarrow{OM}= \overrightarrow{OA}-\overrightarrow{MA}$$
  • $$\displaystyle \overrightarrow{OM}= \dfrac{1}{2}\left ( \overrightarrow{OA}-\overrightarrow{OB} \right )$$
  • $$\displaystyle \overrightarrow{OM}= \dfrac{1}{2}\left ( \overrightarrow{OB}+\overrightarrow{OA} \right )$$
OABC is a tetrahedron express the vectors $$\displaystyle \overrightarrow{BC},\overrightarrow{CA},\:and\:\overrightarrow{AB}$$ in terms of the vectors $$\displaystyle \overrightarrow{OA},\overrightarrow{OB}\:and\:\overrightarrow{OC}$$
  • $$\displaystyle \overrightarrow{BC}= \overrightarrow{OC}+\overrightarrow{OB}\quad;\:\overrightarrow{CA}= \overrightarrow{OA}+\overrightarrow{OC}\quad;\:\overrightarrow{AB}= \overrightarrow{OB}+\overrightarrow{OA}$$
  • $$\displaystyle \overrightarrow{BC}= \overrightarrow{OC}-\overrightarrow{OB}\quad;\:\overrightarrow{CA}= \overrightarrow{OA}-\overrightarrow{OC}\quad;\:\overrightarrow{AB}= \overrightarrow{OB}-\overrightarrow{OA}$$
  • $$\displaystyle \overrightarrow{BC}= \overrightarrow{OC}-\overrightarrow{BO}\quad;\:\overrightarrow{CA}= \overrightarrow{OA}-\overrightarrow{CO}\quad;\:\overrightarrow{AB}= \overrightarrow{OB}-\overrightarrow{AO}$$
  • none of these
Given the vectors $$\bar a$$ and $$\bar b$$ as follows. Find the projections of $$\bar a$$ on $$\bar b$$ and of $$\bar b$$ on $$\bar a$$.
$$\bar a=\hat i+\hat j+\hat k$$; $$\displaystyle \bar b= \sqrt{3}\hat i+3\hat j-2\hat k$$
  • $$\displaystyle \frac{\sqrt{2}+1}{5},\frac{3+\sqrt{3}}{5}.$$
  • $$\displaystyle \frac{\sqrt{3}+1}{4},\frac{3+\sqrt{3}}{3}.$$
  • $$\displaystyle \frac{\sqrt{5}+2}{4},\frac{4+\sqrt{3}}{4}.$$
  • $$\displaystyle \frac{\sqrt{2}+5}{3},\frac{4+\sqrt{3}}{3}.$$
If $$\displaystyle \overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR},$$ then $$\displaystyle P, Q, R$$ are
  • the vertices of an equilateral triangle
  • the vertices of an isosceles triangle
  • collinear
  • None of these
Which of the following statements are correct:
If in triangle OAC, B is the mid-point of AC and $$\displaystyle \overrightarrow{OA}= \vec{a}\:$$ and $$\:\overrightarrow{OB}= \vec{b}$$ then
  • $$\displaystyle \overrightarrow{OC}= \frac{1}{2}\left ( \vec{a}+\vec{b} \right )$$
  • $$\displaystyle \overrightarrow{OC}= 2\vec{b}-2\vec{a}$$
  • $$\displaystyle \overrightarrow{OC}= 2\vec{b}-\vec{a}$$
  • $$\displaystyle \overrightarrow{OC}= 3\vec{a}-2\vec{b}$$
If a, b, c are position vectors of the vertices of a $$\displaystyle \Delta ABC,$$ then $$ \displaystyle \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=$$
  • 0
  • 2a
  • 2b
  • 3c
Projection of the vector $$2i + 3j - 2k$$ on the vector $$i - 2j + 3k$$ is
  • $$\displaystyle 2/\sqrt{\left ( 14 \right )}$$
  • $$\displaystyle 1/\sqrt{\left ( 14 \right )}$$
  • $$\displaystyle 3/\sqrt{\left ( 14 \right )}$$
  • None of these
Given that the vectors $$\bar a,\bar b,\bar c$$ form a base, find the sum of  co-ordinates of the vector: $$3\bar u-\bar v+\bar w$$ 
if $$\bar u=\bar a+\bar c, \bar v=\bar b+\bar c,\bar w=\bar a-\bar b$$;
  • 4
  • 6
  • 2
  • 3
If $$\displaystyle \left ( \bar A+\bar B \right )$$ is perpendicular to$$\bar B$$ and If $$\displaystyle \left ( \bar A+2\bar B \right )$$ is perpendicular to $$\bar A$$, then
  • $$\displaystyle A=\sqrt{2}B$$
  • $$\displaystyle A=2B$$
  • $$\displaystyle 2A=B$$
  • $$\displaystyle A=B$$
$$\displaystyle \overrightarrow{AE}$$
  • $$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a+\bar b \right )$$
  • $$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a-\bar b \right )$$
  • $$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar b-\bar a \right )$$
  • none of these
If $$a$$ is perpendicular to $$b$$ and $$c$$, then
  • $$\displaystyle a\times \left ( b\times c \right )=1$$
  • $$\displaystyle a\times \left ( b\times c \right )=0$$
  • $$\displaystyle a\times \left ( b\times c \right )=-1$$
  • None of these
$$AB=3i+j-k$$ and $$AC=i-j+3k$$. If the point $$P$$ on the line segment $$BC$$ is equidistant from $$AB$$ and $$AC,$$ then $$AP$$ is
  • $$2i-k$$
  • $$i-2k$$
  • $$2i+k$$
  • None of these
The points $$O,A,B,C$$ are the vertices of a pyramid and $$P,Q,R,S$$ are the mid-points of $$OA,OB,BC,AC$$ respectively. If $$\displaystyle \overrightarrow{OA}=a,\overrightarrow{OB}=b,\overrightarrow{OC}=c,$$ express in terms of $$a, b, c$$ the vectors $$\displaystyle \overrightarrow{OP},\overrightarrow{OQ},\overrightarrow{OR}$$ and $$\displaystyle \overrightarrow{OS}$$/
  • $$\displaystyle \overrightarrow{OP}=\dfrac{1}{2}a,$$ $$\overrightarrow{OQ}=\dfrac{1}{2}b,$$ $$\overrightarrow{OR}=\dfrac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\dfrac{1}{2}\left ( a+c \right )$$
  • $$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( a+c \right )$$
  • $$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$
  • $$\displaystyle \overrightarrow{OP}=\frac{1}{2}a,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$
If $$4i+7j+8k, 2i+7j+7k$$ and $$3i+5j+7k$$ are the position vectors of the vertices $$A,B$$ and $$C$$ respectively of triangle $$ABC$$. The position vector of the point where the bisector of angle $$A$$ meets $$BC.$$
  • $$\displaystyle \frac { 1 }{ 3 } \left( 5j+12k \right) $$
  • $$\displaystyle \frac { 1 }{ 3 } \left( 6i+13j+18k \right) $$
  • $$\displaystyle \frac { 2 }{ 3 } \left( 6i+8j+6k \right) $$
  • $$\displaystyle \frac { 2 }{ 3 } \left( -6i-8j-6k \right) $$
Given two vectors $$\displaystyle a=2i-3j+6k$$ on $$\displaystyle b=2i+3j-k$$ and $$\displaystyle \lambda =\frac{the\:projection\:of\:a\:on\:b}{the\:projection\:of\:b\:on\:a},$$ then the value of $$\displaystyle \lambda $$ is
  • 3/7
  • 7/3
  • 3
  • 7
$$\displaystyle \overrightarrow{BC}$$ 
  • $$\displaystyle \overrightarrow{BC}=2(b-a)$$
  • $$\displaystyle \overrightarrow{BC}=2(a-b)$$
  • $$\displaystyle \overrightarrow{BC}=2(b+a)$$
  • none of these
The position vectors of three consecutive vertices of a parallelogram are $$i+j+k, i+3j+5k$$ and $$7i+9j+11k$$. The position vector of the fourth vertex is
  • $$6(i+j+k)$$
  • $$7(i+j+k)$$
  • $$2j-4k$$
  • $$6i+8j+10k$$
Let $$G$$ be the centroid of a triangle $$ABC$$. If $$AB=a,AC=b$$ then the bisector $$AG$$, in terms of vectors $$a$$ and $$b$$ is
  • $$\displaystyle \frac { 2 }{ 3 } \left( a+b \right) $$
  • $$\displaystyle \frac { 1 }{ 6 } \left( a+b \right) $$
  • $$\displaystyle \frac { 1 }{ 3 } \left( a+b \right) $$
  • $$\displaystyle \frac { 1 }{ 2 } \left( a+b \right) $$
What is the maximum number of components into which a vector can be
split ?
  • 2
  • 3
  • 4
  • Infinite
The sum of the three vectors determined by the medians of a triangle directed from the vertices is
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$\displaystyle \frac{1}{3}$$
$$P$$ is any point on the circumcircle of $$\triangle ABC$$ other than the vertices. $$H$$ is the orthocenter of $$\triangle ABC,M$$ is the mid-point of $$PH$$ and $$D$$ is the mid-point of $$BC$$. Then
  • $$AP$$ is opposite side of $$DM$$
  • $$DM$$ is parallel to $$AP$$ 
  • $$DM$$ is perpendicular to $$AP$$
  • None of these
In a parallelogram $$ABCD$$, $$\left| AB \right| =a,\left| AD \right| =b$$ and $$\left| AC \right| =c$$. Then $$DB.AB$$ has the value
  • $$\displaystyle \frac { 3{ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle \frac { { a }^{ 2 }+3{ b }^{ 2 }-{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle \frac { { a }^{ 2 }-{ b }^{ 2 }+3{ c }^{ 2 } }{ 2 } $$
  • $$\displaystyle \frac { { a }^{ 2 }+3{ b }^{ 2 }+{ c }^{ 2 } }{ 2 } $$
If $$S$$ is the circumcentre, $$O$$ is the orthocenter of $$\triangle ABC, $$ then $$SA+SB+SC=$$
  • $$SO$$
  • $$2SO$$
  • $$OS$$
  • $$2OS$$
If $$O$$ and $$O'$$ are circumcenter and orthocenter of a triangle $$ABC$$ then $$\left( OA+OB+OC \right) $$ equals
  • $$2OO'$$
  • $$OO'$$
  • $$O'O$$
  • $$2O'O$$
If $$\displaystyle \vec{a},\vec{b},\vec{c},\vec{d},\vec{e},\vec{f}$$ are position vectors of 6 points A, B, C, D, E & F respectively such that $$\displaystyle 3\vec{a}+4\vec{b}=6\vec{c}+\vec{d}=4\vec{e}+3\vec{f}=\vec{x}$$ then
  • $$\displaystyle \overline{AB}$$ is parallel to $$\displaystyle \overline{CD}$$
  • line AB, CD and EF are concurrent
  • $$\displaystyle \frac{\vec{x}}{7}$$ is position vector of the point dividing CD in ratio 1 : 6
  • A, B, C, D, E & F are coplanar
Any vector in an arbitrary direction can always be replaced by two (or three)
  •  parallel vectors which have the original vector as their resultant.
  •  mutually perpendicular vectors which have the original vector as their resultant.
  •  arbitrary vectors which have the original vector as their resultant.
  •  it is not possible to resolve a vector.
$$12$$ coplanar non collinear forces (all of equal magnitude) maintain a body in equilibrium, then the angle between any two adjacent forces is:
  • $$\;15^{\circ}$$
  • $$\;30^{\circ}$$
  • $$\;45^{\circ}$$
  • $$\;60^{\circ}$$
The value of $$\vec i \times (\vec a \times \vec i) + \vec j \times (\vec a \times \vec j) + \vec k \times (\vec a \times \vec k)$$ is (where $$\vec i, \vec j, \vec k$$ are unit vectors)
  • $$\vec a$$
  • $$2 \vec a$$
  • 0
  • $$- \vec a$$
If a and b are two non-zero and non-collinear vectors, then a + b and a - b are
  • Linearly dependent vectors
  • Linearly dependent and independent vectors
  • Linearly independent vectors
  • None of these
Let $$\vec{a}=\hat{i}+\hat{j}+3\hat{k}\;\&\;\vec{b}=2\hat{i}-3\hat{j}+4\hat{k}$$. If projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\displaystyle\frac{k}{\sqrt{29}}$$, then the value of $$(k-2)$$ is
  • $$9$$
  • $$-9$$
  • $$8$$
  • $$6$$
If the position vectors of the vertices of a triangle be 6i + 4j + 5k, 4i + 5j + 6k and 5i + 6j + 4k then the triangle is
  • Right angled triangle
  • Equilateral triangle
  • Isosceles triangle
  • None of these
In triangle $$ABC$$, which of the following is not true?
429162_24efde2d46de47b4b8b1fe30fba0fdd7.png
  • $$\vec {AB}+\vec {BC}+\vec {CA}=\vec {0}$$
  • $$\vec {AB}+\vec {BC}-\vec {AC}=\vec {0}$$
  • $$\vec {AB}+\vec {BC}-\vec {CA}=\vec {0}$$
  • $$\vec {AB}-\vec {CB}+\vec {CA}=\vec {0}$$
If $$\vec{a}=(\lambda\,x)\hat{i}+(y)\hat{j}+(4z)\hat{k},\,\vec{b}=y\hat{i}+x\hat{j}+3y\hat{k},\,\vec{c}=-z\hat{i}-2z\hat{j}-\begin{pmatrix}(\lambda+1)x\end{pmatrix}\hat{k}$$ are sides of triangle as shown in figure, then value of $$\lambda$$ is (where $$x,\,y,\,z$$ are not all zero)
295574.bmp
  • $$0$$
  • $$2$$
  • $$-1$$
  • $$1$$
The position vectors of $$P$$ and $$Q$$ are respectively $$a$$ and $$b$$. If $$R$$ is a point on $$PQ$$, $$PQ$$ such that $$PR=5PQ$$, then the position vector of $$R$$ is
  • $$5b-4a$$
  • $$5b+4a$$
  • $$4b-5a$$
  • $$4b+5a$$
If $$|\vec {a}| = |\vec {b}| = 1$$ and $$|\vec {a} + \vec {b}| = \sqrt {3}$$, then the value of $$(3\vec {a} - 4\vec {b}) \cdot (2\vec {a} + 5\vec {b})$$ is
  • $$-21$$
  • $$-\dfrac {21}{2}$$
  • $$21$$
  • $$\dfrac {21}{2}$$
If $$\vec { a } \cdot \hat { i } =4$$, then $$\left( \vec { a } \times \hat { j }  \right) \cdot \left( 2\hat { j } -3\hat { k }  \right) $$ is equal to
  • $$12$$
  • $$2$$
  • $$0$$
  • $$-12$$
Let $$ABC$$ be a triangle whose circumcentre is at P.  If the position vectors of $$A, B, C$$ and P are $$\vec {a}, \vec {b}, \vec {c}$$ and $$\dfrac {\vec {a} + \vec {b} + \vec {c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:
  • $$-\left (\dfrac {\vec {a} + \vec {b} + \vec {c}}{2}\right )$$
  • $$\vec {a} + \vec {b} + \vec {c}$$
  • $$\dfrac {(\vec {a} + \vec {b} + \vec {c})}{2}$$
  • $$\vec {0}$$
Let $$\overrightarrow {a} , \overrightarrow {b}$$ and $$\overrightarrow {c}$$ be vectors with magnitudes 3, 4 and 5 respectively and $$\overrightarrow{a} + \overrightarrow {b}+\overrightarrow {c}=\overrightarrow {0}$$, then the value of $$\overrightarrow{a}. \overrightarrow{b}+\overrightarrow{b}. \overrightarrow{c} + \overrightarrow{c}. \overrightarrow{a}$$ is
  • 47
  • 25
  • 50
  • -25
Let $$\vec {a} = \vec {i} + 2\vec {j} + \vec {k}, \vec {b} = \vec {i} - \vec {j} + \vec {k}$$ and $$\vec {c} = \vec {i} + \vec {j} - \vec {k}$$. A vector in the plane of $$\vec {a}$$ and $$\vec {b}$$ has projection $$\dfrac {1}{\sqrt {3}}  \ on\  \vec {c}$$. Then, one such vector is
  • $$4\vec {i} + \vec {j} - 4\vec {k}$$
  • $$3\vec {i} + \vec {j} - 3\vec {k}$$
  • $$4\vec {i} - \vec {j} + 4\vec {k}$$
  • $$2\vec {i} + \vec {j} - 2\vec {k}$$
Find the correct vectorial relationship with the help of the figure above.
493602.jpg
  • $$\vec {x} + \vec {y} = \vec {z}$$
  • $$\vec {y} + \vec {z} = \vec {x}$$
  • $$\vec {x} + \vec {z} = \vec {y}$$
  • $$\vec {z} - \vec {x} = \vec {y}$$
  • $$\vec {z} - \vec {y} = \vec {x}$$
How much does a watch lose per day, if its hands coincide every $$64$$ minutes?
  • $$32\cfrac { 8 }{ 11 } min.$$
  • $$31\cfrac { 8 }{ 11 } min.$$
  • $$32\cfrac { 3 }{ 11 } min.$$
  • $$None\ of\ these$$
$$ABCDEF$$ is a regular hexagon whose centre is $$O$$. The $$\overline { AB } +\overline { AC } +\overline { AD } +\overline { AE } +\overline { AF } $$ is
  • $$2\overline { AO } $$
  • $$3\overline { AO } $$
  • $$5\overline { AO } $$
  • $$6\overline { AO } $$
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