Vector Algebra Test

Question 1
1 / -0

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 $$\lambda=$$

A
$$\displaystyle \frac { 1 }{ 4 } $$

B
$$\displaystyle \frac { 1 }{ 2 } $$

C
$$\displaystyle \frac { 1 }{ 6 } $$

D
$$\displaystyle \frac { 1 }{ 8 } $$

Question 2
1 / -0

If $$a$$ is perpendicular to $$b$$ and $$c$$, then

A
$$\displaystyle a\times \left ( b\times c \right )=1$$

B
$$\displaystyle a\times \left ( b\times c \right )=0$$

C
$$\displaystyle a\times \left ( b\times c \right )=-1$$

D
None of these

Question 3
1 / -0

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 $$\displaystyle \overrightarrow{OS}$$/

A
$$\displaystyle \overrightarrow{OP}=\dfrac{1}{2}a,$$ $$\overrightarrow{OQ}=\dfrac{1}{2}b,$$ $$\overrightarrow{OR}=\dfrac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\dfrac{1}{2}\left ( a+c \right )$$

B
$$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( a+c \right )$$

C
$$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$

D
$$\displaystyle \overrightarrow{OP}=\frac{1}{2}a,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$

Question 4
1 / -0

Given the vectors $$\bar a$$ and $$\bar b$$ as follows. Find the projections of $$\bar a$$ on $$\bar b$$ and of $$\bar b$$ on $$\bar a$$.
$$\bar a=\hat i+\hat j+\hat k$$; $$\displaystyle \bar b= \sqrt{3}\hat i+3\hat j-2\hat k$$

A
$$\displaystyle \frac{\sqrt{2}+1}{5},\frac{3+\sqrt{3}}{5}.$$

B
$$\displaystyle \frac{\sqrt{3}+1}{4},\frac{3+\sqrt{3}}{3}.$$

C
$$\displaystyle \frac{\sqrt{5}+2}{4},\frac{4+\sqrt{3}}{4}.$$

D
$$\displaystyle \frac{\sqrt{2}+5}{3},\frac{4+\sqrt{3}}{3}.$$

Question 5
1 / -0

In a $$\triangle OAB$$, $$L$$ is a point on the side $$AB$$ and $$M$$ is a point on the side $$OB$$ and the lines $$OL$$ and $$AM$$ meet at $$S$$. It is given that $$AS=SM$$ and $$4 OS=3OL$$. and that $$\displaystyle \left (\frac{OM}{OB} \right )=h$$ and $$\displaystyle \left (\frac{AL}{AB} \right )=k.$$ Express the vectors $$\displaystyle \overrightarrow{AM},$$ and $$\displaystyle \overrightarrow{OS}$$ in term of $$a, b$$ and $$h$$ and the vectors $$\displaystyle \overrightarrow{OL}$$ and $$\displaystyle \overrightarrow{OS}$$ in terms of $$a,b$$ and $$k$$ where $$\displaystyle \overrightarrow{OA}=a$$ and $$\displaystyle \overrightarrow{OB}=b.$$ Find $$h$$ and $$k$$

A
$$ \dfrac{1}{2},\dfrac{1}{3}$$

B
$$ \dfrac{1}{3},\dfrac{1}{2}$$

C
$$ \dfrac{1}{6},\dfrac{1}{2}$$

D
$$ \dfrac{1}{3},\dfrac{1}{6}$$

Question 6
1 / -0

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

A
3/7

B
7/3

C
3

D
7

Question 7
1 / -0

Directions For Questions

ABC is a triangle and P,Q are the mid-points of AB, AC respectively. If $$\displaystyle \overrightarrow{AB}=2\vec a$$ and $$\displaystyle \overrightarrow{AC}=2\vec b$$.

...view full instructions

$$\displaystyle \overrightarrow{BC}$$

A
$$\displaystyle \overrightarrow{BC}=2(b-a)$$

B
$$\displaystyle \overrightarrow{BC}=2(a-b)$$

C
$$\displaystyle \overrightarrow{BC}=2(b+a)$$

D
none of these

Question 8
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If $$\displaystyle \overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR},$$ then $$\displaystyle P, Q, R$$ are

A
the vertices of an equilateral triangle

B
the vertices of an isosceles triangle

C
collinear

D
None of these

Question 9
1 / -0

Projection of the vector $$2i + 3j - 2k$$ on the vector $$i - 2j + 3k$$ is

A
$$\displaystyle 2/\sqrt{\left ( 14 \right )}$$

B
$$\displaystyle 1/\sqrt{\left ( 14 \right )}$$

C
$$\displaystyle 3/\sqrt{\left ( 14 \right )}$$

D
None of these

Question 10
1 / -0

Directions For Questions

$$ABCD$$ is parallelogram whose diagonals intersect at $$E$$ and $$M$$ is the mid-Point of $$DC$$. If$$\displaystyle \overrightarrow{AB}=\bar a$$ and $$\displaystyle \overrightarrow{AD}=\bar b,$$express in terms of $$\bar a$$ and $$\bar b$$ the vectors

...view full instructions

$$\displaystyle \overrightarrow{AE}$$

A
$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a+\bar b \right )$$

B
$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a-\bar b \right )$$

C
$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar b-\bar a \right )$$

D
none of these

Submit Test

Vector Algebra ...

Question 1

$$\displaystyle \frac { 1 }{ 4 } $$

$$\displaystyle \frac { 1 }{ 2 } $$

$$\displaystyle \frac { 1 }{ 6 } $$

$$\displaystyle \frac { 1 }{ 8 } $$

Question 2

$$\displaystyle a\times \left ( b\times c \right )=1$$

$$\displaystyle a\times \left ( b\times c \right )=0$$

$$\displaystyle a\times \left ( b\times c \right )=-1$$

None of these

Question 3

$$\displaystyle \overrightarrow{OP}=\dfrac{1}{2}a,$$ $$\overrightarrow{OQ}=\dfrac{1}{2}b,$$ $$\overrightarrow{OR}=\dfrac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\dfrac{1}{2}\left ( a+c \right )$$

$$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( b+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( a+c \right )$$

$$\displaystyle \overrightarrow{OP}=\frac{1}{2}c,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$

$$\displaystyle \overrightarrow{OP}=\frac{1}{2}a,\overrightarrow{OQ}=b,\overrightarrow{OR}=\frac{1}{2}\left ( a+c \right ),\overrightarrow{OS}=\frac{1}{2}\left ( b+c \right )$$

Question 4

$$\displaystyle \frac{\sqrt{2}+1}{5},\frac{3+\sqrt{3}}{5}.$$

$$\displaystyle \frac{\sqrt{3}+1}{4},\frac{3+\sqrt{3}}{3}.$$

$$\displaystyle \frac{\sqrt{5}+2}{4},\frac{4+\sqrt{3}}{4}.$$

$$\displaystyle \frac{\sqrt{2}+5}{3},\frac{4+\sqrt{3}}{3}.$$

Question 5

$$ \dfrac{1}{2},\dfrac{1}{3}$$

$$ \dfrac{1}{3},\dfrac{1}{2}$$

$$ \dfrac{1}{6},\dfrac{1}{2}$$

$$ \dfrac{1}{3},\dfrac{1}{6}$$

Question 6

3/7

7/3

3

7

Question 7

$$\displaystyle \overrightarrow{BC}=2(b-a)$$

$$\displaystyle \overrightarrow{BC}=2(a-b)$$

$$\displaystyle \overrightarrow{BC}=2(b+a)$$

none of these

Question 8

the vertices of an equilateral triangle

the vertices of an isosceles triangle

collinear

None of these

Question 9

$$\displaystyle 2/\sqrt{\left ( 14 \right )}$$

$$\displaystyle 1/\sqrt{\left ( 14 \right )}$$

$$\displaystyle 3/\sqrt{\left ( 14 \right )}$$

None of these

Question 10

$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a+\bar b \right )$$

$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar a-\bar b \right )$$

$$\displaystyle \overrightarrow{AE}=\frac{1}{2}\left ( \bar b-\bar a \right )$$

none of these

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