Write each vector in the form a i + b j.
〈6, -3〉

Verified step by step guidance

1. Understand that the vector given in angle bracket notation 〈x, y〉 can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) is the unit vector in the x-direction and \( \mathbf{j} \) is the unit vector in the y-direction.
2. Identify the components of the vector: here, the x-component is 6 and the y-component is -3.
3. Write the vector as a linear combination of the unit vectors: multiply the x-component by \( \mathbf{i} \) and the y-component by \( \mathbf{j} \).
4. Express the vector as \( 6\mathbf{i} + (-3)\mathbf{j} \).
5. Simplify the expression by writing it as \( 6\mathbf{i} - 3\mathbf{j} \), which is the vector in the form \( a\mathbf{i} + b\mathbf{j} \).

Key Concepts

Vector Representation in Component Form

A vector in two dimensions can be expressed as a combination of its horizontal and vertical components. These components correspond to the vector's projections along the x-axis and y-axis, respectively, and are typically written as a multiple of unit vectors i (x-direction) and j (y-direction).

Unit Vectors i and j

Unit vectors i and j are standard basis vectors in the plane, where i represents a vector of length one in the positive x-direction, and j represents a vector of length one in the positive y-direction. Any 2D vector can be written as a linear combination of i and j.

Converting Coordinate Notation to Vector Form

Given a vector in coordinate form 〈x, y〉, it can be rewritten as x i + y j by multiplying the x-component by i and the y-component by j. This form clearly shows the vector's direction and magnitude along each axis.