Question

Figure 3–39 shows two vectors, A→\overrightarrow{A} and B→\overrightarrow{B}, whose magnitudes are A = 6.8 units and B = 5.5 units. Determine C→\overrightarrow{C} if C→\overrightarrow{C} = A→\overrightarrow{A} + B→\overrightarrow{B}. Give the magnitude and direction for each.

Accepted Answer

Step 1: Understand the problem. You are tasked with finding the resultant vector $$ \vec{C} $$ when $$ \vec{C} = \vec{A} + \vec{B} . $$This involves vector addition, and you need to determine both the magnitude and direction of $$ \vec{C} . $$The magnitudes of $$ \vec{A} $$ and $$ \vec{B} $$ are given as 6.8 units and 5.5 units, respectively. The directions of $$ \vec{A} $$ and $$ \vec{B} $$ are implied in the figure (not provided here). Step 2: Break down the vectors into components. Use trigonometry to resolve $$ \vec{A} $$ and $$ \vec{B} $$ into their x- and y-components. For a vector $$ \vec{V} $$ with magnitude $$ V $$ and angle $$ 	heta  ($$measured from the positive x-axis): $$ V_x = V \cos(	heta) $$ and $$ V_y = V \sin(	heta) . $$Apply this formula to both $$ \vec{A} $$ and $$ \vec{B} . $$Step 3: Add the components of $$ \vec{A} $$ and $$ \vec{B}  to $$find the components of $$ \vec{C} . $$Specifically, $$ C_x = A_x + B_x $$ and $$ C_y = A_y + B_y . $$This step combines the x-components and y-components of the two vectors to form the resultant vector $$ \vec{C} . $$Step 4: Calculate the magnitude of $$ \vec{C} . $$Use the Pythagorean theorem: $$ |\vec{C}| = \sqrt{C_x^2 + C_y^2} . $$This gives the length of the resultant vector. Step 5: Determine the direction of $$ \vec{C} . $$The direction (angle $$ 	heta_C ) $$can be found using the inverse tangent function: $$ 	heta_C = 	an^{-1}(C_y / C_x) . $$Ensure the angle is adjusted based on the quadrant in which $$ \vec{C} $$ lies.

Figure 3–39 shows two vectors, $\vec{A}$ and $\vec{B}$ , whose magnitudes are A = 6.8 units and B = 5.5 units. Determine $\vec{C}$ if $\vec{C}$ = $\vec{A}$ + $\vec{B}$ . Give the magnitude and direction for each.

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Key Concepts

Vector Addition

Magnitude of a Vector

Direction of a Vector