Question

given vector v has magnitude |v| = 86.5 and direction angle θ = 46.7° with respect to the positive x-axis, find the horizontal and vertical components of v.

in the answer box, enter the horizontal and vertical components (both rounded to one decimal place) of v like this:
horizontal = #
vertical = #

Explanation:

Step1: Recall vector component formulas

The horizontal (x) component of a vector \(\mathbf{v}\) with magnitude \(|\mathbf{v}|\) and direction angle \(\theta\) is given by \(v_x = |\mathbf{v}| \cos\theta\), and the vertical (y) component is \(v_y = |\mathbf{v}| \sin\theta\).

Step2: Calculate horizontal component

Given \(|\mathbf{v}| = 86.5\) and \(\theta = 46.7^\circ\), we calculate \(v_x\):
\(v_x = 86.5 \cos(46.7^\circ)\)
Using a calculator, \(\cos(46.7^\circ) \approx 0.686\) (rounded for calculation), so \(v_x \approx 86.5 \times 0.686 \approx 59.4\) (wait, actually, let's use more precise calculation: \(\cos(46.7^\circ)\) is calculated as \(\cos(46.7) \approx \cos(46 + 0.7) = \cos(46^\circ) \cos(0.7^\circ) - \sin(46^\circ) \sin(0.7^\circ)\), but better to use calculator directly. Let's compute \(\cos(46.7^\circ)\): using calculator, \(46.7^\circ\) in radians is \(46.7 \times \frac{\pi}{180} \approx 0.815\) radians. \(\cos(0.815) \approx 0.685\)? Wait, no, let's use calculator input: 86.5 * cos(46.7°). Let's do this: 46.7 degrees, cos(46.7) ≈ 0.6857 (more accurately, using calculator: cos(46.7) ≈ 0.6857). So 86.5 * 0.6857 ≈ 86.5 * 0.6857. Let's calculate 80*0.6857=54.856, 6.5*0.6857=4.45705, total ≈ 54.856 + 4.45705 = 59.31305 ≈ 59.3? Wait, no, maybe my initial approximation was wrong. Wait, let's use calculator properly: 46.7 degrees, cos(46.7) is approximately 0.68568. So 86.5 * 0.68568 = let's compute 86 * 0.68568 = 58.96848, 0.5 * 0.68568 = 0.34284, total = 58.96848 + 0.34284 = 59.31132 ≈ 59.3? Wait, no, maybe I made a mistake. Wait, actually, let's use calculator for cos(46.7°):

Using a calculator (in degree mode), cos(46.7) ≈ 0.685683. Then 86.5 * 0.685683 ≈ 86.5 * 0.685683. Let's compute 86.5 * 0.685683:

86.5 * 0.6 = 51.9

86.5 * 0.08 = 6.92

86.5 * 0.005683 ≈ 86.5 * 0.005 = 0.4325, 86.5 * 0.000683 ≈ 0.0591

So adding up: 51.9 + 6.92 = 58.82 + 0.4325 = 59.2525 + 0.0591 ≈ 59.3116 ≈ 59.3? Wait, but maybe I should use more precise calculation. Wait, no, maybe I messed up the angle. Wait, 46.7 degrees, let's check with calculator:

Using a scientific calculator, enter 46.7, then cos. Let's see, my calculator gives cos(46.7) ≈ 0.685683. So 86.5 * 0.685683 ≈ 59.3 (rounded to one decimal place). Wait, but let's do it more accurately: 86.5 * cos(46.7°). Let's compute 86.5 * cos(46.7) = 86.5 * 0.685683 = let's do 86.5 * 0.685683:

86.5 * 0.685683 = (80 + 6.5) * 0.685683 = 80*0.685683 + 6.5*0.685683 = 54.85464 + 4.4569395 = 59.3115795 ≈ 59.3 (rounded to one decimal place). Wait, but maybe I made a mistake here. Wait, no, let's check the vertical component.

Step3: Calculate vertical component

\(v_y = 86.5 \sin(46.7^\circ)\)
\(\sin(46.7^\circ) \approx 0.7279\) (using calculator: sin(46.7) ≈ 0.7279). So \(v_y = 86.5 \times 0.7279 \approx 86.5 * 0.7279\). Let's calculate 80*0.7279=58.232, 6.5*0.7279=4.73135, total ≈ 58.232 + 4.73135 = 62.96335 ≈ 63.0? Wait, no, let's use precise calculation: sin(46.7°) ≈ 0.72787. So 86.5 * 0.72787 ≈ 86.5 * 0.72787. Let's compute 80*0.72787=58.2296, 6.5*0.72787=4.731155, total ≈ 58.2296 + 4.731155 = 62.960755 ≈ 63.0 (rounded to one decimal place). Wait, but let's check with calculator: 86.5 * sin(46.7°). Sin(46.7) is approximately 0.72787, so 86.5 * 0.72787 ≈ 62.96, which rounds to 63.0. Wait, but maybe my initial cosine calculation was wrong. Wait, let's recheck cos(46.7°):

Using calculator, cos(46.7) = cos(46 + 0.7) = cos(46°)cos(0.7°) - sin(46°)sin(0.7°). Cos(46°) ≈ 0.6947, cos(0.7°) ≈ 0.9999, sin(46°) ≈ 0.7193, sin(0.7°) ≈ 0.0122. So cos(46.7°) ≈ 0.6947*0.9999 - 0.7193*0.0122 ≈ 0.6946 - 0.0088 ≈ 0.6858, which matches the earlier calculation. So 86.5 * 0.6858 ≈ 59.3. And sin(46.7°) ≈ 0.7279, so 86.5 * 0.7279 ≈ 63.0. Wait, but let's use a calculator for precise values:

Using a calculator (like TI-84):

- Calculate horizontal component: 86.5 * cos(46.7°)
- Enter 46.7, press cos: cos(46.7) ≈ 0.685683
- Multiply by 86.5: 86.5 * 0.685683 ≈ 59.3115 ≈ 59.3 (rounded to one decimal place)
- Calculate vertical component: 86.5 * sin(46.7°)
- Enter 46.7, press sin: sin(46.7) ≈ 0.72787
- Multiply by 86.5: 86.5 * 0.72787 ≈ 62.9607 ≈ 63.0 (rounded to one decimal place)

Wait, but let's check with another approach. Maybe I made a mistake in the angle. Wait, the direction angle is with respect to the positive x-axis, so the formulas are correct: \(v_x = |v| \cos\theta\), \(v_y = |v| \sin\theta\). So with \(|\mathbf{v}| = 86.5\) and \(\theta = 46.7^\circ\), the calculations should be correct.

Answer:

horizontal = 59.3
vertical = 63.0