Question

Vector V is 448 m long in a 224° direction. Vector Wis 336 m long in a 75.9° direction. Find the direction!!!! of their vector sum.​

Answer 1

177°

Explanation:

To find the direction of the vector sum of vectors 'V' and 'W', we will first break each vector into its horizontal and vertical components, then sum these components to get the resultant vector, and finally calculate the angle of this resultant vector.

We are given:

• V = 448 m at 224°
• W = 336 m long at 75.9°

`\boxed{\left\begin{array}{ccc}\vec v = < \ v_x, \ v_y > \\\\\text{\underline{Where:}} \\\\ ||\vec v||=√(v_x^2+v_y^2) \\\\ v_x=||\vec v||\cos(\theta)\\\\v_y=||\vec v||\sin(\theta) \\\\ \theta=\tan^(-1)\Big((v_y)/(v_x) \Big) \underbrace{+180 \textdegree}_{\text{if} \ v_x < 0}\end{array}\right}`

Taking vector V and putting it into component form:

`\Longrightarrow \vec V=\big < (448)\cos(224\textdegree),(448)\sin(224\textdegree) \big > \\\\\\\\\therefore \vec V=\big < -322, \ -311\big > \ m`

Now vector W:

`\Longrightarrow \vec W=\big < (336)\cos(75.9\textdegree),(336)\sin(75.9\textdegree) \big > \\\\\\\\\therefore \vec W=\big < 81.9, \ 326\big > \ m`

Now performing the vector sum. Add x-components together and add y-components together:

`\Longrightarrow \vec V + \vec W=\big < -322+81.9, -311+326 \big > \\\\\\\\\therefore \vec V + \vec W=\big < -240, 14.7 \big > \ m`

Now, using the sum of V and W, we can find the direction of this resultant vector:

`\Longrightarrow \theta = \tan^(-1)\left((14.7)/(-240)\right)+ 180 \textdegree\\\\\\\\\Longrightarrow \theta = -3.49 \textdegree+ 180 \textdegree\\\\\\\\\therefore \theta \approx \boxed{177 \textdegree}`

Thus, the resultant direction is approximately 177°.