If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).

Verified step by step guidance

1. First, understand that the problem requires finding the dot product of vector \( u \) with the sum of vectors \( v \) and \( w \). The dot product is defined as \( \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y \) for 2D vectors.
2. Calculate the vector sum \( \mathbf{v} + \mathbf{w} \) by adding their corresponding components: \( (v_x + w_x) \) for the i-component and \( (v_y + w_y) \) for the j-component.
3. Write down the resulting vector from step 2 explicitly as \( \mathbf{v} + \mathbf{w} = (v_x + w_x)\mathbf{i} + (v_y + w_y)\mathbf{j} \).
4. Now, compute the dot product \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \) by multiplying the i-components of \( \mathbf{u} \) and \( \mathbf{v} + \mathbf{w} \), and the j-components of \( \mathbf{u} \) and \( \mathbf{v} + \mathbf{w} \), then summing these products: \( u_x (v_x + w_x) + u_y (v_y + w_y) \).
5. Express the final dot product as a sum of products of components without calculating the numerical value, which completes the setup for the solution.

Key Concepts

Vector Addition

Vector addition involves combining two vectors by adding their corresponding components. For example, if v = i - j and w = 3i - 7j, then v + w = (1+3)i + (-1-7)j = 4i - 8j. This operation is essential before performing the dot product in the given problem.

Dot Product of Vectors

The dot product of two vectors u and v is a scalar calculated by multiplying their corresponding components and summing the results. For vectors u = a1i + b1j and v = a2i + b2j, u ⋅ v = a1a2 + b1b2. This operation measures the extent to which two vectors point in the same direction.

Component Form of Vectors

Vectors can be expressed in component form using unit vectors i and j along the x and y axes, respectively. This form allows easy manipulation of vectors through addition, subtraction, and dot product by working directly with their components.