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Slav-nsk [51]
2 years ago
5

Given three vectors A = 24i + 33j, B = 55i - 12j and C = 2i + 43j (a) Find the magnitude of each vector. (b) Write an expression

for the vector difference A - C. (c) Find the magnitude and direction of the vector difference A-B. (d) In a vector diagram show vector A + B, and A - B, and also show that your diagram agrees qualitatively with your answer.

Physics
1 answer:
slega [8]2 years ago
6 0

Answer:

(a) , .  and .

(b)\vec A - \vec C=22 \hat i -10 \hat j.

(c)|\vec A - \vec B|=63.13 and the direction \theta = 124.56°.

Explanation:

Given that,

,

and

\vec {C}=2 \hat i +43 \hat j

(a) The magnitude of a vector is the square root of the sum of the square of all the components of the vector, i.e. for a ,.

So, the magnitude of the is

|\vec A|=\sqrt {24^2+ 33^2}

\Rightarrow |\vec A|=\sqrt {1665}

.

The magnitude of the is

|\vec B|=\sqrt {55^2+ (-12)^2}

\Rightarrow |\vec B|=\sqrt {3169}

.

And, the magnitude of the is

|\vec C|=\sqrt {2^2+ 43^2}

\Rightarrow |\vec C|=\sqrt {1853}

.

(b) The difference between the two vectors is the difference between the corresponding components of the vectors. So, the required expression of is

\vec A - \vec C=(24 \hat i +33 \hat j) - (2 \hat i +43 \hat j)

\Rightarrow \vec A - \vec C=24 \hat i +33 \hat j - 2 \hat i -43 \hat j

\Rightarrow \vec A - \vec C=22 \hat i -10 \hat j

(c) The expression of is

\vec A - \vec N=(24 \hat i +33 \hat j) - (55 \hat i -12 \hat j)

\Rightarrow \vec A - \vec B=24 \hat i +33 \hat j - 55\hat i +12 \hat j

\Rightarrow \vec A - \vec B=-31 \hat i +45 \hat j\;\cdots (i)

The magnitude of is

|\vec A - \vec B|=\sqrt {(-31)^2+55^2}

\Rightarrow |\vec A - \vec B|=\sqrt {3986}

\Rightarrow |\vec A - \vec B|=63.13

Now, if a vector \vec V= -\alpha \hat i +\beta \hat j in 3rd quadrant having direction \theta with respect to \hat i direction, than

in the anti-clockwise direction.

Here, from equation (i), for the vector \vec A - \vec C, \alpha=31 and \beta=45.

\Rightarrow \theta = \pi-\tan ^{-1}\left(\frac {45}{31}\right)

180°-55.44° [as \pi radian= 180°]

124.56° in the anti-clockwise direction.

(d) Vector diagrams for \vec A +\vec B and \vec A - \vec B has been shown  

in the figure(b) and figure(c) recpectively.

Vector \vec A - \vec B is in 3rd quadrant as calculated in part (c).

While Vector \vec A +\vec B=(24 \hat i +33 \hat j)+(55 \hat i -12 \hat j)

\Rightarrow \vec A +\vec B=79 \hat i +21 \hat j, which is in 1st quadrant as both the components are position has been shown in figure(b).

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