In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j

Verified step by step guidance

1. Identify the components of the vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \). Here, the vector has an \( x \)-component of 1 (from \( \mathbf{i} \)) and a \( y \)-component of -1 (from \( -\mathbf{j} \)). So, \( \mathbf{v} = (1, -1) \).
2. To sketch the vector as a position vector, start at the origin \( (0,0) \) on the coordinate plane.
3. From the origin, move 1 unit in the positive \( x \)-direction (right) and 1 unit in the negative \( y \)-direction (down). Mark the point \( (1, -1) \).
4. Draw an arrow from the origin \( (0,0) \) to the point \( (1, -1) \). This arrow represents the vector \( \mathbf{v} \).
5. To find the magnitude of \( \mathbf{v} \), use the formula for the length of a vector: \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \). Substitute \( x = 1 \) and \( y = -1 \) to get \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).

Key Concepts

Position Vector

A position vector represents the location of a point in space relative to the origin. It is expressed in terms of unit vectors along coordinate axes, such as i and j in two dimensions, indicating horizontal and vertical components respectively.

Vector Components

Vector components break down a vector into its horizontal (i) and vertical (j) parts. For v = i - j, the vector moves one unit right and one unit down, which helps in visualizing and sketching the vector accurately on the coordinate plane.

Magnitude of a Vector

The magnitude of a vector is its length, calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. For v = i - j, magnitude = √(1² + (-1)²) = √2, representing the distance from the origin to the vector's endpoint.