Question

The two vectors a and b in the figure have equal magnitudes of 10.6 m and the angles are ei-34° and 82-10%. Find (a) the x component and (b) the y component of their vector sum ,(e) the magnitude of , and (d) the angle r makes with the positive direction of the x axis. 8, (a) Number Units (b) NumberUnts (c) Number Units (d) Number Units

Answer 1

Using the cosine law for triangles, we have

r² = a² + b² - 2 a b cos (180 - $\theta$₂)

We know that, the angle $\vec{b}$ makes with the +x axis is (34⁰ + 107⁰) = 141⁰

Applying the following relations which are given below as :

a_x = a cos $\theta$ and    a_y = a sin $\theta$

| $\vec{a}$ | = _$\sqrt{}$a_x² + a_y²

$\vec{r}$ = ($\vec{a}$ + $\vec{b}$) = (a_x + b_x) $\hat{i}$ + (a_y + b_y) _$\hat{j}$

(a) The x component of their vector sum ($\vec{r}$) which will be given by -

r_x = (10.6 m) cos 34⁰ + (10.6 m) cos 141⁰

r_x = [(8.78 m) - (8.23 m)]

r_x = 0.55 m

(b) The y component of their vector sum ($\vec{r}$) which will be given by -

r_y = (10.6 m) sin 34⁰ + (10.6 m) sin 141⁰

r_y = [(5.92 m) - (6.67 m)]

r_y = 12.6 m

(c) The magnitude of ($\vec{r}$) which will be given by -

| $\vec{r}$ | = _$\sqrt{}$(0.55 m)² + (12.6 m)²

| $\vec{r}$ | = _$\sqrt{}$159.0625 m²

| $\vec{r}$ | = 12.6 m

(d) The angle ($\vec{r}$) makes with a positive direction of the x-axis which will be given by -

_$\phi$ = tan^-1 [(12.6 m) / (0.55 m)]

_$\phi$ = tan^-1 (22.9)

_$\phi$ = 87.5 degree