Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.
u • v - u • w

Verified step by step guidance

1. Identify the vectors: \( \mathbf{u} = \langle -2, 1 \rangle \), \( \mathbf{v} = \langle 3, 4 \rangle \), and \( \mathbf{w} = \langle -5, 12 \rangle \).
2. Calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \) using the formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \).
3. Calculate the dot product \( \mathbf{u} \cdot \mathbf{w} \) using the formula \( \mathbf{u} \cdot \mathbf{w} = u_1w_1 + u_2w_2 \).
4. Subtract the result of \( \mathbf{u} \cdot \mathbf{w} \) from \( \mathbf{u} \cdot \mathbf{v} \).
5. The expression \( \mathbf{u} \cdot \mathbf{v} - \mathbf{u} \cdot \mathbf{w} \) is evaluated by performing the subtraction from the previous step.

Key Concepts

Dot Product of Vectors

The dot product is a fundamental operation in vector algebra that combines two vectors to produce a scalar. For vectors u = 〈u1, u2〉 and v = 〈v1, v2〉, the dot product is calculated as u • v = u1*v1 + u2*v2. This operation is useful for determining the angle between vectors and for projecting one vector onto another.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, resulting in a new vector. For vectors u and v, the subtraction is defined as u - v = 〈u1 - v1, u2 - v2〉. This concept is essential for understanding how vectors interact and can be used in various applications, including physics and engineering.

Scalar Operations

Scalar operations involve arithmetic with scalar quantities, which are single numerical values. In the context of vectors, scalar operations can include addition, subtraction, and multiplication of the results from vector operations like the dot product. Understanding how to manipulate scalars is crucial for solving expressions that involve vectors.