Given vectors u and v, find: v - 3u. 
u = 〈-1, 2〉, v = 〈3, 0〉

Verified step by step guidance

1. Identify the components of vector \( u \) as \( u = \langle -1, 2 \rangle \) and vector \( v \) as \( v = \langle 3, 0 \rangle \).
2. To find \( v - 3u \), first calculate \( 3u \) by multiplying each component of \( u \) by 3: \( 3u = 3 \times \langle -1, 2 \rangle = \langle 3 \times -1, 3 \times 2 \rangle \).
3. Simplify the expression for \( 3u \) to get \( 3u = \langle -3, 6 \rangle \).
4. Subtract \( 3u \) from \( v \) by subtracting the corresponding components: \( v - 3u = \langle 3, 0 \rangle - \langle -3, 6 \rangle \).
5. Perform the subtraction for each component: \( v - 3u = \langle 3 - (-3), 0 - 6 \rangle \).

Key Concepts

Vector Operations

Vector operations involve mathematical procedures applied to vectors, such as addition, subtraction, and scalar multiplication. In this case, we are performing a subtraction of a scaled vector from another vector. Understanding how to manipulate vectors is essential for solving problems involving them.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a single number), which scales the vector's magnitude without changing its direction. For example, multiplying vector u by -3 scales its components, affecting the resulting vector's position in the coordinate system.

Coordinate Representation of Vectors

Vectors can be represented in a coordinate system using ordered pairs or tuples, such as u = 〈-1, 2〉 and v = 〈3, 0〉. Each component corresponds to a dimension in the space, allowing for geometric interpretation and algebraic manipulation of vectors in two-dimensional space.