Vector Algebra Test

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Vector Algebra Test - 18

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Question 1
1 / -0

Find the vector $$w$$ with the initial point $$(4,1,2)$$ and final point $$(1,6,5)$$.

A
$$(3,-5,-3)$$

B
$$(0,7,3)$$

C
$$(-3,5,3)$$

D
None of these

Solution

Given points
$$A(4,1,2)$$ and $$B(1,6,5)$$
$$\vec{OA}=4\hat{i}+\hat{j}+2\hat{k}$$
$$\vec{OB}=\hat{i}+6\hat{j}+5\hat{k}$$
vector
$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{AB}=\hat{i}+6\hat{j}+5\hat{k}-(4\hat{i}+\hat{j}+2\hat{k})$$
$$\vec{AB}=\hat{i}+6\hat{j}+5\hat{k}-4\hat{i}-\hat{j}-2\hat{k}$$
$$\vec{AB}=-3\hat{i}+5\hat{j}+3\hat{k}$$
$$AB=(-3,5,3)$$
Question 2
1 / -0

Find $$2v$$, when $$u=(3,4,-2)$$ and $$v=(0,-4,0)$$.

A
$$(0,-8,0)$$

B
$$(3,8,0)$$

C
$$(3,-8,-2)$$

D
None of these

Solution

Given points
$$u(3,4,-2)$$ and $$v(0,-4,0)$$
$$\vec{u}=3\hat{i}+4\hat{j}-2\hat{k}$$
$$\vec{v}=-4\hat{j}$$
vector
$$2\vec{v}=2\times -4\vec{j}$$
$$2\vec{v}=-8\vec{j}=(0,-8,0)$$
Question 3
1 / -0

Find the magnitude of the vector which joins the point $$A$$$$(4,5,6)$$ to $$B$$$$(10,11,12)$$.

A
$$12.29$$

B
$$10.39$$

C
$$10.29$$

D
None of these

Solution

Given points
$$A(4,5,6)$$ and $$B(10,11,12)$$
$$\vec{OA}=4\hat{i}+5\hat{j}+6\hat{k}$$
$$\vec{OB}=10\hat{i}+11\hat{j}+12\hat{k}$$
vector
$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{AB}=10\hat{i}+11\hat{j}+12\hat{k}-(4\hat{i}+5\hat{j}+6\hat{k})$$
$$\vec{AB}=10\hat{i}+11\hat{j}+12\hat{k}-4\hat{i}-5\hat{j}-6\hat{k}$$
$$\vec{AB}=6\hat{i}+6\hat{j}+6\hat{k}$$
magnitude of AB
$$\left | \vec{AB} \right |=\sqrt{6^2+6^2+6^2}$$

$$\left | \vec{AB} \right |=\sqrt{36+36+36}$$

$$\left | \vec{AB} \right |=\sqrt{36\times 3}$$

$$\left | \vec{AB} \right |=6\sqrt{3}$$

$$\left | \vec{AB} \right |=6\times 1.732$$

$$\left | \vec{AB} \right |=10.39$$
Question 4
1 / -0

Which of the following is not a unit vector for all values of $$\theta$$?

A
$$(\cos\theta)i - (\sin\theta)j$$

B
$$(\sin\theta)i + (\cos\theta)j$$

C
$$(\sin2\theta)i - (\cos\theta)j$$

D
$$(\cos2\theta)i - (\sin2\theta)j$$

Solution

Let $$n$$ be any vector.
Then $$n$$ is a unit vector, if $$||n||=1$$
We know that, $$\cos^2 \theta + \sin^2 \theta=1$$
For $$n=(\sin 2\theta)i - (\cos \theta)j$$
$$||n||=\sin^{2} 2\theta + \cos^{2} \theta \neq 1$$
Hence, $$(\sin^{2} 2\theta)i + (\cos^{2} \theta)j$$ is not a unit vector.
Question 5
1 / -0

Find the vector $$w$$ with the initial point $$(8,-10,3)$$ and final point $$(1,10,7)$$.

A
$$(7,10,-4)$$

B
$$(-4,10,4)$$

C
$$(-7,20,4)$$

D
None of these

Solution

Given points
$$A(8,-10,3)$$ and $$B(1,10,7)$$
$$\vec{OA}=8\hat{i}-10\hat{j}+3\hat{k}$$
$$\vec{OB}=\hat{i}+10\hat{j}+7\hat{k}$$
vector w from A to B
$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{w}=\hat{i}+10\hat{j}+7\hat{k}-(8\hat{i}-10\hat{j}+3\hat{k})$$
$$\vec{w}=\hat{i}+10\hat{j}+7\hat{k}-8\hat{i}+10\hat{j}-3\hat{k}$$
$$\vec{AB}=-7\hat{i}+20\hat{j}+4\hat{k}=(-7,20,4)$$
Question 6
1 / -0

The vector joining vector $$\vec{A}$$ to $$\vec{B}$$ is represented by:

A
$$\vec{A}-\vec{B}$$

B
$$\vec{B}-\vec{A}$$

C
$$\vec{A}+\vec{B}$$

D
None of these

Solution

Given vectors
$$\vec{A}$$ and $$\vec{B}$$
$$\vec{AB}=\vec{B}-\vec{A}$$
Question 7
1 / -0

Find the magnitude of the vector which joins the point $$A$$$$(-1,-3,-1)$$ to $$B$$$$(0,0,0)$$.

A
$$\sqrt{21}$$

B
$$\sqrt{10}$$

C
$$\sqrt{11}$$

D
None of these

Solution

Given points
$$A(-1,-3,-1)$$ and $$B(0,0,0)$$
$$\vec{OA}=-\hat{i}-3\hat{j}-\hat{k}$$
$$\vec{OB}=0$$
vector AB from A to B
$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{AB}=0-(-\hat{i}-3\hat{j}-\hat{k})$$
$$\vec{AB}=+\hat{i}+3\hat{j}+\hat{k})$$
magnitude of AB
$$\left | \vec{AB} \right |=\sqrt{1^2+3^2+1^2}$$

$$\left | \vec{AB} \right |=\sqrt{1+9+1}$$

$$\left | \vec{AB} \right |=\sqrt{11}$$
Question 8
1 / -0

Find the vector $$w$$ with the initial point $$(a,b,c)$$ and final point $$(a+1,b+2,c+3)$$.

A
$$(a,b+2,c+3)$$

B
$$(a-1,b-2,c-3)$$

C
$$(1,2,3)$$

D
None of these

Solution

Given points
$$A(a,b,c)$$ and $$B(a+1,b+2,c+3)$$
$$\vec{OA}=a\hat{i}+b\hat{j}+c\hat{k}$$
$$\vec{OB}=(a+1)\hat{i}+(b+2)\hat{j}+(c+3)\hat{j}$$
vector $$w$$ from $$A$$ to $$B$$
$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{w}=(a+1)\hat{i}+(b+2)\hat{j}+(c+3)\hat{j}-(a\hat{i}+b\hat{j}+c\hat{k})$$
$$\vec{w}=a\hat{i}+b\hat{j}+c\hat{k}+\hat{i}+2\hat{j}+3\hat{j}-a\hat{i}-b\hat{j}-c\hat{k}$$
$$\vec{w}=\hat{i}+2\hat{j}+3\hat{j}=(1,2,3)$$
Question 9
1 / -0

Find the magnitude of the vector which joins the point $$A$$$$(a,2,c)$$ to $$B$$$$(a+1,5,c+3)$$.

A
$$\sqrt{18}$$

B
$$\sqrt{21}$$

C
$$\sqrt{20}$$

D
None of these

Solution

Given points
$$A(a,2,c)$$ and $$B(a+1,5,c+3)$$
$$\vec{OA}=a\hat{i}+2\hat{j}+c\hat{k}$$
$$\vec{OB}=(a+1)\hat{i}+5\hat{j}+(c+3)\hat{j}$$

$$\vec{AB}=\vec{OB}-\vec{OA}$$
$$\vec{AB}=(a+1)\hat{i}+5\hat{j}+(c+3)\hat{j}-(a\hat{i}+2\hat{j}+c\hat{k})$$
$$\vec{AB}=a\hat{i}+5\hat{j}+c\hat{k}+\hat{i}+3\hat{j}-a\hat{i}-2\hat{j}-c\hat{k}$$
$$\vec{AB}=\hat{i}+3\hat{j}+3\hat{j}$$

magnitude of AB
$$\left | \vec{AB} \right |=\sqrt{1^2+3^2+3^2}$$

$$\left | \vec{AB} \right |=\sqrt{1+9+9}$$

$$\left | \vec{AB} \right |=\sqrt{19}$$
none of option is matched
Question 10
1 / -0

The coordinates of a point in $$3$$-D space is $$(3,1,2)$$. Then the position vector of the point is:

A
$$3\hat i + 3\hat j + 3\hat k$$

B
$$3\hat i + 2\hat j + \hat k$$

C
$$3\hat i + \hat j + 2\hat k$$

D
$$\hat i + 2\hat j + 3\hat k$$

Solution

A position vector is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference.
Given point is $$(3,1,2)$$ in $$3$$-D space.
$$x,y$$ and $$z$$ coordinates are represented by vectors $$i,j$$ and $$k$$ respectively.
Hence, position vector is given by $$3\hat i+\hat j+2\hat k$$.

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Vector Algebra Test - 18

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Vector Algebra Test - 18

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Question 1

$$(3,-5,-3)$$

$$(0,7,3)$$

$$(-3,5,3)$$

None of these

Solution

Question 2

$$(0,-8,0)$$

$$(3,8,0)$$

$$(3,-8,-2)$$

None of these

Solution

Question 3

$$12.29$$

$$10.39$$

$$10.29$$

None of these

Solution

Question 4

$$(\cos\theta)i - (\sin\theta)j$$

$$(\sin\theta)i + (\cos\theta)j$$

$$(\sin2\theta)i - (\cos\theta)j$$

$$(\cos2\theta)i - (\sin2\theta)j$$

Solution

Question 5

$$(7,10,-4)$$

$$(-4,10,4)$$

$$(-7,20,4)$$

None of these

Solution

Question 6

$$\vec{A}-\vec{B}$$

$$\vec{B}-\vec{A}$$

$$\vec{A}+\vec{B}$$

None of these

Solution

Question 7

$$\sqrt{21}$$

$$\sqrt{10}$$

$$\sqrt{11}$$

None of these

Solution

Question 8

$$(a,b+2,c+3)$$

$$(a-1,b-2,c-3)$$

$$(1,2,3)$$

None of these

Solution

Question 9

$$\sqrt{18}$$

$$\sqrt{21}$$

$$\sqrt{20}$$

None of these

Solution

Question 10

$$3\hat i + 3\hat j + 3\hat k$$

$$3\hat i + 2\hat j + \hat k$$

$$3\hat i + \hat j + 2\hat k$$

$$\hat i + 2\hat j + 3\hat k$$

Solution

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