CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 7 - MCQExams.com

If $$2\overrightarrow { a }\cdot \overrightarrow { b } =\left| \overrightarrow { a }  \right| \left| \overrightarrow { b }  \right| $$ then the angle between $$\vec { a }$$ and $$ \vec { b }$$ is 
  • $${30}^{o}$$
  • $${0}^{o}$$
  • $${90}^{o}$$
  • $${60}^{o}$$
For non-zero vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ if $$|\overrightarrow{a}+\overrightarrow{b}| < |\overrightarrow{a}-\overrightarrow{b}|$$, then $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ are
  • Collinear
  • Perpendicular to each other
  • Inclined at an acute angle
  • Inclined at an obtuse angle
Let $$\vec {a} = i + 2j + k, \vec {b} = i - j + k$$ and $$\vec {c} = i + j - k$$, a vector in the plane $$\vec {a}$$ and $$\vec {b}$$ whose projection on $$\vec {c}$$ is $$\dfrac {1}{\sqrt {3}}$$ is _____
  • $$3i + j - 3k$$
  • $$4i + j - 4k$$
  • $$i + j - 2k$$
  • $$2i + j + 2k$$
The value of $$x$$ if $$x(\hat { i } +\hat { j } +\hat { k } )$$ is a unit vector is
  • $$\pm \cfrac { 1 }{ \sqrt { 3 } } $$
  • $$\pm \sqrt { 3 } $$
  • $$\pm 3$$
  • $$\pm \cfrac{1}{3}$$
If $$\vec { a }$$ and $$\vec { b }$$ are two unit vector and $$\theta$$ is the angle between them, then $$\left( \vec { a } +\vec { b }  \right)$$ is a unit vector if $$\theta =$$
  • $$\dfrac { \pi }{ 3 } $$
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { 2\pi }{ 3 } $$
The position vector of a point which is $$2$$ units away from $$3\hat i + 4\hat j + 2\hat k$$ along the $$x$$-axis is:
  • $$\hat i + 4\hat j + 2\hat k$$
  • $$3\hat i + 2\hat j + 2\hat k$$
  • $$5\hat i + 4\hat j + 2\hat k$$
  • $$3\hat i + 4\hat j + \hat k$$
The projection of $$\vec {i} - \vec {j}$$ on $$z$$-axis is
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
The vector $$b = 3j + 4k$$ is to be written as the sum of a vector $$b_{1}$$ parallel to $$a = i + j$$ and a vector $$b_{2}$$ perpendicular to $$a$$. Then $$b_{1}$$ is equal to
  • $$\dfrac {3}{2}(i + j)$$
  • $$\dfrac {2}{3}(i + j)$$
  • $$\dfrac {1}{2}(i + j)$$
  • $$\dfrac {1}{3}(i + j)$$
If $$\vec a, \vec b$$ and $$\vec c$$ are three non-coplanar vectors, then $$(\vec a +\vec  b -\vec  c) \cdot [(\vec a - \vec b) \times (\vec b - \vec c)$$ equals
  • $$\vec 0$$
  • $$\vec a.(\vec b\times \vec c)$$
  • $$\vec a.(\vec c\times \vec b)$$
  • $$3\vec a.(\vec b\times \vec c)$$
If $$\overrightarrow { a } $$ is vector of magnitude $$x$$ , $$m$$ is non-zero scalar and $$m\overrightarrow { a } $$ is a unit vector then x in terms of m is:
  • $$m=x$$
  • $$x=\left| m \right| $$
  • $$x=\cfrac { 1 }{ \left| m \right| } $$
  • $$x=m/2$$
If $$\vec { a } ,\vec { b } ,\vec { c } $$ are mutually perpendicular unit vectors, then $$\left| \vec { a } +\vec { b } +\vec { c }  \right| $$ is equal to
  • $$3$$
  • $$\sqrt { 3 } $$
  • Zero
  • $$1$$
If $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$ are the position vectors of the points B and C respectively, then the position vector of the point D such that $$\overrightarrow{BD}=4\overrightarrow{BC}$$ is
  • $$4(\overrightarrow{c}-\overrightarrow{b})$$
  • $$-4(\overrightarrow{c}-\overrightarrow{b})$$
  • $$4\overrightarrow{c}-3\overrightarrow{b}$$
  • $$4\overrightarrow{c}+3\overrightarrow{b}$$
Consider the vectors $$\bar{a}=\hat{i}-2\hat{j}+\hat{k}$$ and $$ \bar{b}=4\hat{i}-4\hat{j}+7\hat{k}$$
Find the scalar projection of $$\bar{a}$$ on $$\bar{b}.$$
  • 1
  • $$\dfrac{19}{9}$$
  • $$\dfrac{17}{9}$$
  • $$\dfrac{23}{9}$$
The magnitude of the scalar $$p$$ for which the vector $$p\left( -3\hat { i } -2\hat { j } +13\hat { k }  \right) $$ is of unit length is:
  • $$\dfrac{1}{8}$$
  • $$\dfrac{1}{64}$$
  • $$\sqrt { 182 } $$
  • $$\dfrac{1}{\sqrt { 182 }} $$
Let $$\vec{v_1}, \vec{v_2}, \vec{v_3} , \vec{v_4} $$ be unit vectors in the xy - plane, one each in the interior of the four quadrants. Which of the following statements is necesserily true?
  • $$\vec{v_1}, \vec{v_2}, \vec{v_3} , \vec{v_4} = 0$$
  • There exist i, j with $$ 1 \le i \le j \le 4 $$ such that $$ \vec{v_i} + \vec{v_j} $$ is in the first quadrant.
  • There exist i, j with $$ 1 \le i \le j \le 4 $$ such that $$ \vec{v_i} + \vec{v_j} < 0$$
  • There exist i, j with $$ 1 \le i \le j \le 4 $$ such that $$ \vec{v_i} + \vec{v_j} >0 $$
If the position vector $$\overrightarrow{a}$$ of the point $$(5, n)$$ is such that $$|\overrightarrow{a}|=13$$, then the value/values of n be
  • $$\pm 8$$
  • $$\pm 12$$
  • 8 only
  • 12 only
If $$\overrightarrow a$$ and $$\overrightarrow b$$ are unit vectors, then angle between $$\overrightarrow a$$ and $$\overrightarrow b$$ for $$\sqrt 3 \overrightarrow a - \overrightarrow b$$ to be unit vector is
  • $$60^o$$
  • $$45^o$$
  • $$30^o$$
  • $$90^o$$
If $$a\times b=c$$, $$b\times c=a$$ and $$a,b,c$$ be the mod of the vectors $$a,b,c$$ respectively, then
  • $$a=1, b=1$$
  • $$c=1, a=1$$
  • $$a\cdot \left( b\times c \right) =1$$
  • $$b=1,c=a$$
Let $$u, v$$ and $$w$$ be such that $$\left| u \right| =1, \left| v \right| =3$$ and $$\left| w \right| =2$$. If the projection of $$v$$ along $$u$$ is equal to that of $$w$$ along $$u$$ and vectors $$v$$ and $$w$$ are perpendicular to each other, then $$\left| u-v+w \right| $$ equals
  • $$2$$
  • $$\sqrt { 7 } $$
  • $$\sqrt { 14 } $$
  • $$14$$
A vector $$R$$ is given by $$R=A\times \left( B\times C \right) $$, which of the following is true?
  • $$R$$ must be perpendicular to $$B$$
  • $$R$$ is parallel to $$A$$
  • $$R$$ must be parallel to $$B$$
  • None of the above
If $$a=\hat{i}+\hat{j}, b=2\hat{j}-\hat{k}$$ and $$r\times a=b\times a, r\times b=a\times b$$, then a unit vector in the direction of $${r}$$ is?
  • $$\displaystyle\frac{1}{\sqrt{11}}(\hat{i}+3\hat{j}-\hat{k})$$
  • $$\displaystyle \frac{1}{\sqrt{11}}(\hat{i}-3\hat{j}+\hat{k})$$
  • $$\displaystyle \frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$$
  • None of these
The resultant of $$P$$ and $$Q$$ is $$R$$. If $$Q$$ is doubled, $$R$$ is also doubled and if $$Q$$ is reversed, $$R$$ is again doubled. Then, $$P^{2} : Q^{2} : R^{2}$$ given by
  • $$2 : 2 : 3$$
  • $$3 : 2 : 2$$
  • $$2 : 3 : 2$$
  • $$2 : 3 : 1$$
If the scalar projection of the vector $$xi-j+k$$ on the vector $$2i-j+5k$$ is $$\dfrac { 1 }{ \sqrt { 30 }  } $$ then value of $$x$$ is equal to
  • $$\dfrac { -5 }{ 2 } $$
  • $$6$$
  • $$-6$$
  • $$3$$
In the given diagram, if PQ$$=$$A, QR$$=$$B and RS$$=$$C, then PS will be equal to : 
671333_36bf027e8880479cb3f3b590b9e93fca.jpg
  • $$A-B+C$$
  • $$A+B-C$$
  • $$A+B+C$$
  • $$A-B-C$$
  • $$-A-B-C$$
Let $$\square PQRS$$ be a quadrilateral. If $$M$$ and $$N$$ are midpoints of the sides $$PQ$$ and $$RS$$ respectively then $$\overline {PS} + \overline {QR} =$$
  • $$3\overline {MN}$$
  • $$4\overline {MN}$$
  • $$2\overline {MN}$$
  • $$2\overline {NM}$$
If $$\vec{a}$$ and $$\vec{b}$$ are the vectors determined by two adjacent sides of regular hexagon, then vector $$EF$$ is
  • $$(\vec{a} + \vec{b})$$
  • $$(\vec{a} - \vec{b})$$
  • $$2 \vec{a}$$
  • $$2\vec{b}$$
If $$p=\hat {i} + \hat{j}, q=4\hat{k}-\hat{j}$$ and $$r=\hat{i}+\hat{k}$$, then the unit vector in the direction of $$3p+q-2r$$ is
  • $$\dfrac{1}{3}(\hat{i}+2\hat{j}+2\hat{k})$$
  • $$\dfrac{1}{3}(\hat{i}-2\hat{j}-2\hat{k})$$
  • $$\dfrac{1}{3}(\hat{i}-2\hat{j}+2\hat{k})$$
  • $$\hat{i}-2\hat{j}+2\hat{k}$$
If $$a.b=0$$ and $$a+b$$ makes an angle of $${ 60 }^{ o }$$ with $$b$$, then $$\left| a \right| $$ is equal to
  • $$0$$
  • $$\cfrac { 1 }{ \sqrt { 3 } } \left| b \right| $$
  • $$\cfrac { 1 }{ \left| b \right| } $$
  • $$\left| b \right| $$
  • $$\sqrt { 3 } \left| b \right| $$
If $$a\cdot b =0$$ and $$a + b$$ makes an angle $$60^o$$ with $$a$$, then
  • $$|a| = 2 |b|$$
  • $$2 |a| = |b|$$
  • $$|a| = \sqrt 3 |b|$$
  • $$|a| = |b|$$
  • $$\sqrt 3 |a | = |b|$$
Let $$ABCD$$ be a parallelogram. If $$AB=\hat { i } +3\hat { j } +7\hat { k } , AD=2\hat { i } +3\hat { j } -5\hat { k } $$ and $$p$$ is a unit vector parallel to $$AC$$, then $$p$$ is equal to
  • $$\dfrac { 1 }{ 3 } \left( 2\hat { i } +\hat { j } +2\hat { k } \right) $$
  • $$\dfrac { 1 }{ 3 } \left( 2\hat { i } +2\hat { j } +2\hat { k } \right) $$
  • $$\dfrac { 1 }{ 7 } \left( 3\hat { i } +6\hat { j } +2\hat { k } \right) $$
  • $$\dfrac { 1 }{ 7 } \left( 6\hat { i } +2\hat { j } +3\hat { k } \right) $$
  • $$\dfrac { 1 }{ 7 } \left( 6\hat { i } +2\hat { j } -3\hat { k } \right) $$
Let $$u, v$$ and $$w$$ be vectors such that $$u + v + w = 0$$. If $$|u| = 3, |v| = 4$$ and $$|w| = 5$$, then $$u . v + v . w + w.u$$ is equal to
  • $$0$$
  • $$-25$$
  • $$25$$
  • $$50$$
  • $$47$$
Let $$P\left( 1,2,3 \right) $$ and $$Q\left( -1,-2,-3 \right) $$ be the two points and let $$O$$ be the origin. Then, $$\left| PQ+OP \right| $$ is equal to
  • $$\sqrt { 13 } $$
  • $$\sqrt { 14 } $$
  • $$\sqrt { 24 } $$
  • $$\sqrt { 12 } $$
  • $$\sqrt { 8 } $$
If $$\widehat i + \widehat j, \widehat j + \widehat k, \widehat i + \widehat k$$ are the position vectors of the vertices of a $$\Delta ABC$$ taken in order, then $$\angle A$$ is equal to
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{5}$$
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{3}$$
The angle between the two vectors $$\hat { i } +\hat { j } +\hat { k }$$ and $$ 2\hat { i } -2\hat { j } +2\hat { k } $$ is equal to
  • $$\cos ^{ -1 }{ \left( \dfrac { 2 }{ 3 } \right) } $$
  • $$\cos ^{ -1 }{ \left( \dfrac { 1 }{ 6 } \right) } $$
  • $$\cos ^{ -1 }{ \left( \dfrac { 5 }{ 6 } \right) } $$
  • $$\cos ^{ -1 }{ \left( \dfrac { 1 }{ 18 } \right) } $$
  • $$\cos ^{ -1 }{ \left( \dfrac { 1 }{ 3 } \right) } $$
If $$a=\hat { i } +\hat { j } +\hat { k } $$, $$b=4\hat { i } +3\hat { j } +4\hat { k }$$ and $$c=\hat { i } +\alpha \hat { j } +\beta \hat { k } $$ are coplanar and $$\left| c \right| =\sqrt { 3 } $$, then
  • $$\alpha =\sqrt { 2 } ,\beta =1$$
  • $$\alpha =1,\beta =\pm 1$$
  • $$\alpha =\pm 1,\beta =1$$
  • $$\alpha =\pm 1,\beta =-1$$
  • $$\alpha =-1,\beta =\pm 1$$
If $$\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} , |\vec{b}| = 5 $$ and the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$ \dfrac{\pi}{6} $$, then the area of the triangle formed by these two vectors as two sides is 
  • $$\dfrac{15}{4}$$
  • $$\dfrac{15}{2}$$
  • $$15$$
  • $$\dfrac{15\sqrt 3}{2}$$
  • $$15 \sqrt 3$$
If $$\lambda (3 \widehat i + 2 \widehat j - 6 \widehat k)$$ is a unit vector, then the values of $$\lambda $$ are
  • $$\pm \dfrac{1}{7}$$
  • $$\pm 7$$
  • $$\pm \sqrt{43}$$
  • $$\pm \dfrac{1}{\sqrt{43}}$$
  • $$\pm \dfrac{1}{\sqrt{}7}$$
Let $$a,b,c$$ be three non-zero vectors such that no two of these are collinear. If the vectors $$a+2b$$ is collinear with $$c$$ and $$b+3c$$ is collinear with $$a$$ ($$\lambda$$ being some non-zero scalar), then $$a+2b+6c$$ equals to
  • $$\lambda a$$
  • $$\lambda b$$
  • $$\lambda c$$
  • $$0$$
Let $$a,b,c$$ be three non-zero vectors such that $$a+b+c=0$$, then $$\lambda b\times a+b\times c+c\times a=0$$, where $$\lambda$$ is
  • $$1$$
  • $$2$$
  • $$-1$$
  • $$-2$$
Let $$\vec { a } =\hat { i } +\hat { j } -\hat { k } ,\vec { b } =\hat { i } -\hat { j } +\hat { k } $$ and $$\vec { c } $$ be a unit vector perpendicular to $$\vec { a } $$ and coplanar with $$\vec { a } $$ and $$\vec { b } $$, then $$\vec { c } $$ is
  • $$\cfrac { 1 }{ \sqrt { 2 } } \left( \hat { j } +\hat { k } \right) $$
  • $$\cfrac { 1 }{ \sqrt { 2 } } \left( \hat { j } -\hat { k } \right) $$
  • $$\cfrac { 1 }{ \sqrt { 6 } } \left( \hat { i } -2\hat { j } +\hat { k } \right) $$
  • $$\cfrac { 1 }{ \sqrt { 6 } } \left( 2\hat { i } -\hat { j } +\hat { k } \right) $$
If $$a=2i+2j+3k, b=-1+2j+k$$ and $$c = 3i + j$$, then $$a+tb$$ this perpendicular to $$c$$; if $$t$$ is equal to 
  • 2
  • 4
  • 6
  • 8
If $$3p + 2q =i + j + k$$ and $$3p - 2q = i - j - k$$, then the angle between $$p$$ and $$q$$ is
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{2}$$
  • $$\pi$$
If $$\overline{a},\overline{b},\overline{c},$$ are unit vectors such that $$\overline{a}+\overline{b}+\overline{c}+\overline{c.a}=$$
  • $$\dfrac{3}{2}$$
  • $$-\dfrac{3}{2}$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
If the scalar projection of the vectors $$xi-j+k$$ on the vector $$2i-j+5k$$ is $$\cfrac { 1 }{ \sqrt { 30 }  } $$, then the value of $$x$$ is equal to 
  • $$-\cfrac { 5 }{ 2 } $$
  • $$6$$
  • $$-6$$
  • $$3$$
$$a$$ and $$c$$ are unit vectors and $$|b| = 4$$. If angle between $$b$$ and $$c$$ is $$\cos^{-1}\left (\dfrac {1}{4}\right )$$ and $$a\times b = 2a\times c$$, then $$b= \lambda a + 2c$$, where $$\lambda$$ is equal to
  • $$\pm \dfrac {1}{4}$$
  • $$\pm \dfrac {1}{2}$$
  • $$\pm 4$$
  • None of the above
Let $$\vec{OB} =\hat { i } +2\hat { j } +2\hat { k }$$ and $$\vec{OA} =4\hat { i } +2\hat { j } +2\hat { k } $$. The distance of the point $$B$$ from the straight line passing through $$A$$ and parallel to the vector $$2\hat { i } +3\hat { j } +6\hat { k } $$ is
  • $$\dfrac { 7\sqrt { 5 } }{ 9 } $$
  • $$\dfrac { 5\sqrt { 7 } }{ 9 } $$
  • $$\dfrac { 3\sqrt { 5 } }{ 7 } $$
  • $$\dfrac { 9\sqrt { 5 } }{ 7 } $$
  • $$\dfrac { 9\sqrt { 7 } }{ 5 } $$
Given that A$$+$$B$$+$$C$$=0$$. Out of three vectors, two are equal in magnitude and the magnitude of third vector is $$\sqrt{2}$$ times that of either of the two having equal magnitude. Then, the angles between the vectors are given by.
  • $$30^o, 60^o, 90^o$$
  • $$45^o, 45^o, 90^o$$
  • $$45^o, 60^o, 90^o$$
  • $$90^o, 135^o, 135^o$$
The vectors $$ 2\hat{i} + 3\hat{j}$$, $$5\hat{i}+ 6\hat{j }$$ and $$8\hat{i }+ \lambda\hat{j }$$have their initial points at ( 1, 1 ) . Find the value of $$\lambda $$ so that the vectors terminate on one straight line.
  • 9
  • 8
  • 7
  • 6
If $$\overline {e} = l\overline {i} + m\overline {j} + n\overline {k}$$ is a unit vector, the maximum value of $$lm + mn + nl$$ is
  • $$-\dfrac {1}{2}$$
  • $$0$$
  • $$1$$
  • $$\dfrac {3}{2}$$
If $$|\vec {a}| = 3, |\vec {b}| = 4$$ and $$|\vec {a} - \vec {b}| = 7$$ then $$|\vec {a} + \vec {b}| =$$
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
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