Question

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

Accepted Answer

Identify the components of vectors $$ \mathbf{u} $$ and $$ \mathbf{v} $$ from the figure. Typically, each vector can be broken down into its horizontal (x) and vertical (y) components. For example, $$ \mathbf{u} = (u_x, u_y) $$ and $$ \mathbf{v} = (v_x, v_y) . $$Write down the components of $$ \mathbf{u} $$ and $$ \mathbf{v} $$ explicitly. If the figure provides magnitudes and directions, use trigonometric functions to find components: $$ u_x = |\mathbf{u}| \cos 	heta_u $$, $$ u_y = |\mathbf{u}| \sin 	heta_u $$, and similarly for $$ \mathbf{v} . $$Add the corresponding components of the two vectors to find the components of $$ \mathbf{u} + \mathbf{v} $$: $$ (u_x + v_x, u_y + v_y) . $$This is the vector sum in component form. Express the resulting vector $$ \mathbf{u} + \mathbf{v}  in $$vector notation as $$ \langle u_x + v_x, u_y + v_y angle . $$This notation clearly shows the horizontal and vertical components of the sum. Optionally, if needed, find the magnitude and direction of $$ \mathbf{u} + \mathbf{v} $$ using the formulas: magnitude $$ = \sqrt{(u_x + v_x)^2 + (u_y + v_y)^2} $$ and direction $$ = 	an^{-1} \left( \frac{u_y + v_y}{u_x + v_x} ight) .$$

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

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Key Concepts

Vector Addition

Vector Notation

Component-wise Addition of Vectors