Answer
R = 7.8 KM
θ = 38°
Work Step by Step
The components of vector A ⃗ ̅ are the projections of the vector in the x-axis and the y-axis.
If 𝜃 is defined as the angle between a vector A ⃗ ̅ and the + x-axis the components of the vector A ⃗ ̅ are as follows :
$ A_{x}$= A cos 𝜃 -- (1)
$A_{y}$= A sin 𝜃 --(2)
So, The Magnitude of a vector can be found by using the Pythagoras theorem :
A= $|A| ⃗$ = $\sqrt A_x^2 + A_y^2$
If follows that agents of the angel $\theta $is found by dividing $A_{y}$ by $A_{x}$.
$ tan \theta$= $\frac{Ay}{Ax}$
$\theta$ = $tan^{-1} \frac{Ay}{Ax}$
So, In the question, we have given some information which are as follows :
A ⃗ ̅ =2.6 KM, Due North
B ⃗ ̅ = 4 KM, Due East
C ⃗ ̅ = 3.1 KMDue NorthEast
To Find: We have to find the magnitude and the direction of resultant displacement R ⃗ ̅.
And,
We also asked to show that R ⃗ ̅ in the diagram in Fig 1 which agrees qualitatively with our answer.
So Now A ⃗ ̅ has only on the component in the + y-directions and B ⃗ ̅ has only one component in the + x-directions,
we represent vector C ⃗ ̅ in the form of its x and y components using Equations (1) and (2) :
Cx= $C cos 45^{\circ}$=(3.1 KM) ($cos 45^{\circ}$)
=2.2 KM
Cy= $C cos 45^{\circ}$=(3.1 KM) ($cos 45^{\circ}$)
=2.2 KM
So the components of all the three victories are as follows :
Vector x components y Components
A ⃗ ̅ 0 2,6 KM
B ⃗ ̅ 4 KM 0
C ⃗ ̅ 2.2 Km 2.2 Km
Rx=6.2 KM Ry=4.8 Km
Thus, the X and Y components of resultant displacement are as follows :
Rx=6.2 KM Ry=4.8 Km
The magnitude of the resultant displacement is found from the equation (1)
R=$ \sqrt R_x^2 + R_y^2$
=$\sqrt (6.2KM)^{2}+(4.8KM)^{2}$
=7.8 Km
The angle made by resultant displacement with the x-axis is found from the equation (4)
$\theta$= $tan^(-1)$\frac{4.8 KM}{6.2 KM}
= $38^{\circ}$
Thus, The calculation of $\theta$ shows that R lies in the first quadrant.
Therefore diagram in figure 1 agrees qualitatively with the result in the diagram in figure 1 shows qualitatively with the result.
