Answer:
(a) , . and .
(b)
.
(c)
and the direction
124.56°.
Explanation:
Given that,
,
and
![\vec {C}=2 \hat i +43 \hat j](https://tex.z-dn.net/?f=%5Cvec%20%7BC%7D%3D2%20%5Chat%20i%20%2B43%20%5Chat%20j)
(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}](https://tex.z-dn.net/?f=%7C%5Cvec%20A%7C%3D%5Csqrt%20%7B24%5E2%2B%2033%5E2%7D)
![\Rightarrow |\vec A|=\sqrt {1665}](https://tex.z-dn.net/?f=%5CRightarrow%20%7C%5Cvec%20A%7C%3D%5Csqrt%20%7B1665%7D)
.
The magnitude of the is
![|\vec B|=\sqrt {55^2+ (-12)^2}](https://tex.z-dn.net/?f=%7C%5Cvec%20B%7C%3D%5Csqrt%20%7B55%5E2%2B%20%28-12%29%5E2%7D)
![\Rightarrow |\vec B|=\sqrt {3169}](https://tex.z-dn.net/?f=%5CRightarrow%20%7C%5Cvec%20B%7C%3D%5Csqrt%20%7B3169%7D)
.
And, the magnitude of the is
![|\vec C|=\sqrt {2^2+ 43^2}](https://tex.z-dn.net/?f=%7C%5Cvec%20C%7C%3D%5Csqrt%20%7B2%5E2%2B%2043%5E2%7D)
![\Rightarrow |\vec C|=\sqrt {1853}](https://tex.z-dn.net/?f=%5CRightarrow%20%7C%5Cvec%20C%7C%3D%5Csqrt%20%7B1853%7D)
.
(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)](https://tex.z-dn.net/?f=%5Cvec%20A%20-%20%5Cvec%20C%3D%2824%20%5Chat%20i%20%2B33%20%5Chat%20j%29%20-%20%282%20%5Chat%20i%20%2B43%20%5Chat%20j%29)
![\Rightarrow \vec A - \vec C=24 \hat i +33 \hat j - 2 \hat i -43 \hat j](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cvec%20A%20-%20%5Cvec%20C%3D24%20%5Chat%20i%20%2B33%20%5Chat%20j%20-%202%20%5Chat%20i%20-43%20%5Chat%20j)
![\Rightarrow \vec A - \vec C=22 \hat i -10 \hat j](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cvec%20A%20-%20%5Cvec%20C%3D22%20%5Chat%20i%20-10%20%5Chat%20j)
(c) The expression of is
![\vec A - \vec N=(24 \hat i +33 \hat j) - (55 \hat i -12 \hat j)](https://tex.z-dn.net/?f=%5Cvec%20A%20-%20%5Cvec%20N%3D%2824%20%5Chat%20i%20%2B33%20%5Chat%20j%29%20-%20%2855%20%5Chat%20i%20-12%20%5Chat%20j%29)
![\Rightarrow \vec A - \vec B=24 \hat i +33 \hat j - 55\hat i +12 \hat j](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cvec%20A%20-%20%5Cvec%20B%3D24%20%5Chat%20i%20%2B33%20%5Chat%20j%20-%2055%5Chat%20i%20%2B12%20%5Chat%20j)
![\Rightarrow \vec A - \vec B=-31 \hat i +45 \hat j\;\cdots (i)](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cvec%20A%20-%20%5Cvec%20B%3D-31%20%5Chat%20i%20%2B45%20%5Chat%20j%5C%3B%5Ccdots%20%28i%29)
The magnitude of is
![|\vec A - \vec B|=\sqrt {(-31)^2+55^2}](https://tex.z-dn.net/?f=%7C%5Cvec%20A%20-%20%5Cvec%20B%7C%3D%5Csqrt%20%7B%28-31%29%5E2%2B55%5E2%7D)
![\Rightarrow |\vec A - \vec B|=\sqrt {3986}](https://tex.z-dn.net/?f=%5CRightarrow%20%7C%5Cvec%20A%20-%20%5Cvec%20B%7C%3D%5Csqrt%20%7B3986%7D)
![\Rightarrow |\vec A - \vec B|=63.13](https://tex.z-dn.net/?f=%5CRightarrow%20%7C%5Cvec%20A%20-%20%5Cvec%20B%7C%3D63.13)
Now, if a vector
in 3rd quadrant having direction
with respect to
direction, than
in the anti-clockwise direction.
Here, from equation (i), for the vector
,
and
.
![\Rightarrow \theta = \pi-\tan ^{-1}\left(\frac {45}{31}\right)](https://tex.z-dn.net/?f=%5CRightarrow%20%5Ctheta%20%3D%20%5Cpi-%5Ctan%20%5E%7B-1%7D%5Cleft%28%5Cfrac%20%7B45%7D%7B31%7D%5Cright%29)
180°-55.44° [as \pi radian= 180°]
124.56° in the anti-clockwise direction.
(d) Vector diagrams for
and
has been shown
in the figure(b) and figure(c) recpectively.
Vector
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)](https://tex.z-dn.net/?f=%5Cvec%20A%20%2B%5Cvec%20B%3D%2824%20%5Chat%20i%20%2B33%20%5Chat%20j%29%2B%2855%20%5Chat%20i%20-12%20%5Chat%20j%29)
, which is in 1st quadrant as both the components are position has been shown in figure(b).