Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v

Verified step by step guidance

1. Identify the vector \( \mathbf{v} = \langle 4, 3 \rangle \) and the scalar \( -5 \).
2. Recall that scalar multiplication of a vector involves multiplying each component of the vector by the scalar.
3. Multiply the x-component of \( \mathbf{v} \) by \( -5 \): calculate \( -5 \times 4 \).
4. Multiply the y-component of \( \mathbf{v} \) by \( -5 \): calculate \( -5 \times 3 \).
5. Combine the results to write the new vector as \( \langle -5 \times 4, -5 \times 3 \rangle \).

Key Concepts

Vector Representation

Vectors are quantities defined by both magnitude and direction, often represented as ordered pairs or components in coordinate form, such as u = 〈x, y〉. Understanding how vectors are written and interpreted is essential for performing operations like addition, scalar multiplication, and more.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the vector's magnitude and possibly its direction (if the scalar is negative), but not its direction relative to the scalar's sign. For example, multiplying v = 〈4, 3〉 by -5 results in 〈-20, -15〉.

Component-wise Operations

Vector operations like addition, subtraction, and scalar multiplication are performed component-wise, meaning each corresponding component is operated on individually. This approach simplifies calculations and helps visualize vector transformations in the coordinate plane.