question completion status:\n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20\n21 22 23 24 25 26 27 28 29…

question completion status:\n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20\n21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40\n41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60\nquestion 7\n1.6 points save answer\ntwo displacement vectors, \\(\\vec{a}\\) and \\(\\vec{b}\\), are shown in the figure. the magnitudes of the displacements are \\(a = 10.0\\,\\text{m}\\) and \\(b = 5.00\\,\\text{m}\\).\nwhat is the magnitude of resultant \\(\\vec{r}\\), such that \\(\\vec{r} = \\vec{a} + \\vec{b}\\)?\nhint: review section 1.8 in textbook (component method of vector addition), example 9 page 16 in textbook, and example 1-2 from our lecture video.\nfigure of vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) with angles: \\(\\vec{a}\\) makes 42.0° with +y towards -x, \\(\\vec{b}\\) makes 31.0° with -x towards -y\noptions:\n- 15.0 m\n- 9.00 m\n- 12.0 m\n- 14.0 m

question completion status:\n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20\n21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40\n41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60\nquestion 7\n1.6 points save answer\ntwo displacement vectors, \\(\\vec{a}\\) and \\(\\vec{b}\\), are shown in the figure. the magnitudes of the displacements are \\(a = 10.0\\,\\text{m}\\) and \\(b = 5.00\\,\\text{m}\\).\nwhat is the magnitude of resultant \\(\\vec{r}\\), such that \\(\\vec{r} = \\vec{a} + \\vec{b}\\)?\nhint: review section 1.8 in textbook (component method of vector addition), example 9 page 16 in textbook, and example 1-2 from our lecture video.\nfigure of vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) with angles: \\(\\vec{a}\\) makes 42.0° with +y towards -x, \\(\\vec{b}\\) makes 31.0° with -x towards -y\noptions:\n- 15.0 m\n- 9.00 m\n- 12.0 m\n- 14.0 m

Answer

Explanation:

Step1: Find x-components of vectors

For (\vec{A}): angle with +y is (42.0^\circ), so x-component (A_x = -A\sin(42.0^\circ)) (negative as it's in -x direction). (A = 10.0) m, so (A_x = -10.0\sin(42.0^\circ) \approx -10.0\times0.6691 \approx -6.691) m.
For (\vec{B}): angle with -x is (31.0^\circ), so x-component (B_x = -B\cos(31.0^\circ)) (negative as it's in -x direction). (B = 5.00) m, so (B_x = -5.00\cos(31.0^\circ) \approx -5.00\times0.8572 \approx -4.286) m.
Total x-component: (R_x = A_x + B_x \approx -6.691 - 4.286 \approx -10.977) m.

Step2: Find y-components of vectors

For (\vec{A}): y-component (A_y = A\cos(42.0^\circ)) (positive as it's in +y direction). (A_y = 10.0\cos(42.0^\circ) \approx 10.0\times0.7431 \approx 7.431) m.
For (\vec{B}): angle with -x is (31.0^\circ), so y-component (B_y = -B\sin(31.0^\circ)) (negative as it's in -y direction). (B_y = -5.00\sin(31.0^\circ) \approx -5.00\times0.5150 \approx -2.575) m.
Total y-component: (R_y = A_y + B_y \approx 7.431 - 2.575 \approx 4.856) m.

Step3: Calculate magnitude of (\vec{R})

Use (R = \sqrt{R_x^2 + R_y^2}).
(R_x \approx -10.977), (R_y \approx 4.856).
(R = \sqrt{(-10.977)^2 + (4.856)^2} \approx \sqrt{120.5 + 23.58} \approx \sqrt{144.08} \approx 12.0) m.

Answer:

12.0 m