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The x and y-components of vector A are 4...

The x and y-components of vector A are 4 m and 6 m respectively. The x and y-components of vector A + B are 10 m and 9 m respectively. Calculate for the vector B the following:
(a) its x and y-components
(b) its length
(c ) the angle it makes with x-axis.

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To solve the problem step-by-step, we will calculate the components of vector B given the components of vector A and the resultant vector A + B. ### Given: - The x-component of vector A, \( A_x = 4 \, \text{m} \) - The y-component of vector A, \( A_y = 6 \, \text{m} \) - The x-component of vector A + B, \( R_x = 10 \, \text{m} \) - The y-component of vector A + B, \( R_y = 9 \, \text{m} \) ### Step 1: Calculate the x-component of vector B Using the equation for the x-component of the resultant vector: \[ R_x = A_x + B_x \] Substituting the known values: \[ 10 = 4 + B_x \] To find \( B_x \), rearrange the equation: \[ B_x = 10 - 4 = 6 \, \text{m} \] ### Step 2: Calculate the y-component of vector B Using the equation for the y-component of the resultant vector: \[ R_y = A_y + B_y \] Substituting the known values: \[ 9 = 6 + B_y \] To find \( B_y \), rearrange the equation: \[ B_y = 9 - 6 = 3 \, \text{m} \] ### Step 3: Calculate the length of vector B The length (magnitude) of vector B can be calculated using the Pythagorean theorem: \[ |B| = \sqrt{B_x^2 + B_y^2} \] Substituting the values of \( B_x \) and \( B_y \): \[ |B| = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \, \text{m} \approx 6.71 \, \text{m} \] ### Step 4: Calculate the angle that vector B makes with the x-axis The angle \( \theta \) can be calculated using the tangent function: \[ \tan(\theta) = \frac{B_y}{B_x} \] Substituting the values: \[ \tan(\theta) = \frac{3}{6} = 0.5 \] To find \( \theta \), take the arctangent: \[ \theta = \tan^{-1}(0.5) \approx 26.57^\circ \] ### Summary of Results: (a) The x and y-components of vector B are: - \( B_x = 6 \, \text{m} \) - \( B_y = 3 \, \text{m} \) (b) The length of vector B is: - \( |B| \approx 6.71 \, \text{m} \) (c) The angle that vector B makes with the x-axis is: - \( \theta \approx 26.57^\circ \)

To solve the problem step-by-step, we will calculate the components of vector B given the components of vector A and the resultant vector A + B. ### Given: - The x-component of vector A, \( A_x = 4 \, \text{m} \) - The y-component of vector A, \( A_y = 6 \, \text{m} \) - The x-component of vector A + B, \( R_x = 10 \, \text{m} \) - The y-component of vector A + B, \( R_y = 9 \, \text{m} \) ...
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