Question

Vector A⃗ points in the negative y direction and has a magnitude of 7 units. Vector B⃗ has twice the magnitude and points in the positive x direction.
Find the direction and magnitude of A⃗ +B⃗ .
Express your answer as a whole number.

Answer 1

So vector A has a magnitude of 7 units in the negative Y direction and the Vector B has a magnitude of 14 units in the positive x. Which means that A has a coordinate of A(0,-7) nad B is B(14,0).

-The direction and the magnitude of A+B is -7 down and 14 forward

Answer 2

The magnitude of the vector $(\vec A+\vec B)$ is $\boxed{16 \text{ units}}$ and the direction of the vector $(\vec A+\vec B)$ is $\boxed{26.56^\circ\text{ anticlockwise}}$

Further Explanation:

$\Vec A$ points in the negative Y-direction and has a magnitude of $7\text{ units}$, so we will show it as a line of length $7\text{ units}$ in negative Y-direction.

$\Vec B$ has twice the magnitude and points in the positive X-direction, so we will show it by a line of double the length as before in positive X-direction, as shown in the figure below.

According to Cartesian coordinate system, the resultant will start either from tail of $\vec A$ and ends at head of $\vec B$ and vice-versa.

Now, resultant will be given by  

$\begin{aligned}\vec R&=\vec A+\vec B\\\vec R&=({-7\hat j+14\hat i})\\\end{aligned}$

Hence, the magnitude of resultant or of vector $\vec A + \vec B$ is given by:

$\begin{aligned}|{\vec R}|&=\sqrt{({7^2}+{14^2})}\\&=\sqrt{(49+196)}\\&=\sqrt{245}\\&=15.65{\text{ units}}\\\end{aligned}$

It can be expressed in whole number as:

$\boxed{|\vec R|=16\text{ units}}$

Now, the direction of the resultant vector $\vec R$ or of the vector $\vec A + \vec B$ is given by:

$\tan\theta=\dfrac{{|{\vec A}|}}{{|{\vec B}|}}$

Now, substituting the values of vector $\vec A$ and $\vec B$:

$\begin{aligned}\tan\theta&=\dfrac{7}{{14}}\\\tan\theta&=0.5\\\theta&={\tan^{-1}}(0.5)\\&=\boxed{26.56^\circ}\\\end{aligned}$

Thus, the magnitude of the vector $(\vec A+\vec B)$ is $\boxed{16 \text{ units}}$ and the direction of the vector $(\vec A+\vec B)$ is $\boxed{26.56^\circ\text{ anticlockwise}}$

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Answer Details:

Grade: High School

Subject: Physics

Chapter: Vectors and Scalars

Keywords:

Vector, scalars, vector A, Vector B, addition, resultant, unit, magnitude, direction, positive, X-axis, Y- axis, negative, 7 units, 14 units, twice the magnitude.