Chapter 1: Problem 75

Sketch the vectors with the components \(\vec{A}=\left(A_{x}, A_{y}\right)=\) \((30.0 \mathrm{~m},-50.0 \mathrm{~m})\) and \(\vec{B}=\left(B_{x}, B_{y}\right)=(-30.0 \mathrm{~m}, 50.0 \mathrm{~m}),\) and find the magnitudes of these vectors.

Short Answer

Expert verified

Question: Sketch the vectors A with components (30.0, -50.0) m and B with components (-30.0, 50.0) m, and find their magnitudes. Answer: The vectors A and B are sketched with components (30.0, -50.0) m and (-30.0, 50.0) m, respectively. The magnitudes of both vectors A and B are found to be 58.3 m.

Step by step solution

Reading the vector components

First, let's read the components of the two vectors A and B: - Vector A has components \(A_x = 30.0\) m and \(A_y = -50.0\) m, so vector A can be written as \(\vec{A} = (30.0, -50.0)\) m. - Vector B has components \(B_x = -30.0\) m and \(B_y = 50.0\) m, so vector B can be written as \(\vec{B} = (-30.0, 50.0)\) m.

Sketching the vectors

Using a Cartesian coordinate system, we can now sketch the two vectors with the given components. - To sketch \(\vec{A}\), start at the origin and draw an arrow that goes 30.0 m to the right and 50.0 m downward from the origin. This represents vector A with components \((30.0, -50.0)\) m. - To sketch \(\vec{B}\), start at the origin and draw an arrow that goes 30.0 m to the left and 50.0 m upward from the origin. This represents vector B with components \((-30.0, 50.0)\) m.

Finding the magnitudes of the vectors

Now we can find the magnitudes of the vectors using the formula \(\| \vec{A} \| = \sqrt{A_x^2 + A_y^2}\) for a vector \(\vec{A}=(A_x, A_y)\). - For vector \(\vec{A}\), the magnitude is: \(\| \vec{A} \| = \sqrt{(30.0 \mathrm{~m})^2 + (-50.0 \mathrm{~m})^2} = \sqrt{900 \mathrm{~m}^2 + 2500 \mathrm{~m}^2} = \sqrt{3400 \mathrm{~m}^2} = 58.3 \mathrm{~m}\) - Similarly, for vector \(\vec{B}\), the magnitude is: \(\| \vec{B} \| = \sqrt{(-30.0 \mathrm{~m})^2 + (50.0 \mathrm{~m})^2} = \sqrt{900 \mathrm{~m}^2 + 2500 \mathrm{~m}^2} = \sqrt{3400 \mathrm{~m}^2} = 58.3 \mathrm{~m}\) So, the magnitudes of vectors A and B are both 58.3 m.

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Sketch the vectors with the components \(\vec{A}=\left(A_{x}, A_{y}\right)=\) \((30.0 \mathrm{~m},-50.0 \mathrm{~m})\) and \(\vec{B}=\left(B_{x}, B_{y}\right)=(-30.0 \mathrm{~m}, 50.0 \mathrm{~m}),\) and find the magnitudes of these vectors.

Short Answer

Step by step solution

Reading the vector components

Sketching the vectors

Finding the magnitudes of the vectors

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