Question

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.

Answer 1

(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°.

Step-by-step 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).