Vector Algebra Test

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Vector Algebra Test - 73

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Question 1
1 / -0

Let $$\vec{a}=\hat{i}-\hat{j},\vec{b}=\hat{i}-\hat{j}=\vec{c}=\hat{i}-\hat{j}$$, if $$\vec{d}$$ is a unit vector such that $$\vec{a}.\vec{d}=0=|\vec{b}\vec{c}\vec{d}|$$ then $$\vec{d}$$ equals:

A
$$\pm\dfrac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}$$

B
$$\pm\dfrac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{3}}$$

C
$$\pm\dfrac{\hat{i}+\hat{j}+2\hat{k}}{\sqrt{3}}$$

D
$$\pm\hat{k}$$

Solution
Question 2
1 / -0

If $$\vec a = 4 i + 5 j - k , \vec { b } = i - 4 j + 5 k , \vec c = 3 i + j - k$$ such that $$\vec { p } \perp \vec { a } , \vec { p } \perp \overline { b }$$ and $$\vec { p . c } = 21 ,$$ then $$p =$$

A
$6$( i + j + k )$$

B
$7$( i + j + k )$$

C
$6$( i - j - k )$$

D
$7$( i - j - k )$$

Solution
Question 3
1 / -0

Vector $$\vec { x }$$ satisfying the relation $$\vec { A } . \overline { x } = c$$ and $$\vec { A } \times \vec { x } = \vec { B }$$ is

A
$$\frac { c \vec { A } - ( \overline { A } \times \vec { B } ) } { | \vec { A } | }$$

B
$$\frac { c \vec { A } - ( \vec { A } \times \vec { B } ) } { | \vec { A } | ^ { 2 } }$$

C
$$\frac { c \vec { A } + ( \vec { A } \times \vec { B } ) } { | \vec { A } | ^ { 2 } }$$

D
None

Solution
Question 4
1 / -0

For any vector $$\vec a$$, the value of $${ (\vec a\times \hat i) }^{ 2 }+{ (\vec a\times \hat j) }^{ 2 }+{ (\vec a\times \hat k) }^{ 2 }$$ is equal to

A
$$4{ |\vec a| }^{ 2 }$$

B
$$2{ |\vec a| }^{ 2 }$$

C
$${|\vec a |}^{ 2 }$$

D
$$3{ |\vec a| }^{ 2 }$$

Solution
Question 5
1 / -0

In a parallelogram ABD, $$|\overset { \_ }{ A\overset { \rightarrow }{ B } } |=a,|\overset { \_ }{ A\overset { \rightarrow }{ D } } |=b$$ and $$|\overset { \_ \\ \quad \quad \rightarrow }{ AC } |=c,$$, $$\overset { \_ \rightarrow }{ AB } .\overset { \_ \rightarrow }{ DB } $$ has the value :

A
$$\frac { 1 }{ 2 } ({ 3a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 })$$

B
$$\frac { 1 }{ 2 } ({ a }^{ 2 }-{ b }^{ 2 }+{ c }^{ 2 })$$

C
$$\frac { 1 }{ 2 } ({ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 })$$

D
$$\frac { 1 }{ 3 } ({ b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 })$$

Solution
Question 6
1 / -0

If the position vectors of the vertices $$A,B$$ and $$C$$ of a $$\Delta ABC$$ are respectively $$4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$$ and $$2\hat{i}+5\hat{j}+7\hat{k}$$, then the position vector of the point, where the bisector of $$\angle A$$ meets $$BC$$ is:

A
$$\frac{1}{2}(4\hat{i}+8\hat{j}+11\hat{k})$$

B
$$\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$$

C
$$\frac{1}{4}(8\hat{i}+14\hat{j}+19\hat{k})$$

D
$$\frac{1}{3}(6\hat{i}+8\hat{j}+15\hat{k})$$
Question 7
1 / -0

let $$\bar { a } $$ be a unit vector and $$\bar { b } $$ be a non-zero vector not parallel to $$\bar { a } $$ if two sides of a triangle are represented by the vectors $$\sqrt { 3 } \left( \bar { a } \times \bar { b } \right) \quad and\quad \bar { b } - \left( \bar { a } .\bar { b } \right) \bar { a } $$ then the angles of triangle are

A
$$90^{ \circ },{ 60 }^{ \circ },{ 30 }^{ \circ }$$

B
$$45^{ \circ },{ 45 }^{ \circ },{ 90 }^{ \circ }$$

C
$$60^{ \circ },{ 60 }^{ \circ },{ 60 }^{ \circ }$$

D
$$75^{ \circ },{ 45 }^{ \circ },{ 60 }^{ \circ }$$

Solution
Question 8
1 / -0

In a triangle ABC, if $$A=(0, 0), B=(3, 3\sqrt{3}), C=(-3\sqrt{3}, 3)$$ then the vector of magnitude $$2\sqrt{2}$$ units directed along $$\overline{AO}$$, where O is the circumcentre of triangle ABC is?

A
$$(1-\sqrt{3})\bar{i}+(1+\sqrt{3})\bar{j}$$

B
$$\sqrt{3}\bar{i}+2\bar{j}$$

C
$$\bar{i}-\sqrt{3}\bar{j}$$

D
$$\bar{i}+2\bar{j}$$

Solution
Question 9
1 / -0

If $$a=i-j,b=i+j,c=i+3j+5k$$ and $$n$$ is a unit vector such that $$b,n=0,a,n=0$$ then the value of $$|c,n|$$ is equal to

A
$$1$$

B
$$3$$

C
$$5$$

D
$$2$$

Solution

$$\begin{array}{l} { b_{ n } }=0 \\ \left( { \hat { i } +\hat { j } } \right) \left( { a\hat { i } +\hat { i } e } \right) =0 \\ a+b=c \\ \left( { -\hat { j } } \right) \left( { a\hat { i } +b\hat { j } } \right) =0 \\ a-b=0\, \, \, \, \, \, \, a=0=b \\ \left( { \bar { c } \cdot \bar { n } } \right) \\ \left( { \hat { i } +3\hat { j } +5\hat { b } } \right) \cdot \left( { c\hat { k } } \right) \\ =5c\, =\sqrt { { a^{ 2 } }+{ b^{ 2 } }+{ c^{ 2 } } } =1 \\ =5 \end{array}$$
Question 10
1 / -0

$$If\quad |\overrightarrow { a } |$$ =2 and $$\quad |\overrightarrow { b } |$$=3 and $$\quad |\overrightarrow { a } |$$.$$\quad |\overrightarrow { b } |$$ =0. Then (a x (a x (a x (a x b)))) is equal to

A
$$4\hat { b } $$

B
$$-4\hat { b }$$

C
$$4\hat { a } $$

D
$$-4\hat { a } $$

Solution