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

Verified step by step guidance

1. Step 1: Understand the vector notation. The vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) is given in terms of the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) represents the unit vector in the x-direction and \( \mathbf{j} \) represents the unit vector in the y-direction.
2. Step 2: Sketch the vector. To sketch \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), start at the origin (0,0) on a coordinate plane. Move 1 unit in the positive x-direction (right) and 1 unit in the negative y-direction (down). Draw an arrow from the origin to this point (1, -1).
3. Step 3: Set up the formula for the magnitude of a vector. The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) is given by \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \).
4. Step 4: Substitute the components of the vector into the magnitude formula. For \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), the components are \( a = 1 \) and \( b = -1 \). Substitute these values into the formula: \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).
5. Step 5: Simplify the expression under the square root. Calculate \( 1^2 + (-1)^2 \) to find the expression under the square root, which will give you the magnitude of the vector.

Key Concepts

Position Vectors

A position vector represents a point in space relative to an origin. In a two-dimensional Cartesian coordinate system, a position vector can be expressed in terms of its components along the x-axis and y-axis, typically denoted as v = xi + yj, where i and j are the unit vectors in the x and y directions, respectively.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector v = xi + yj, the magnitude is given by |v| = √(x² + y²). This value represents the distance from the origin to the point defined by the vector in the coordinate system.

Unit Vectors

Unit vectors are vectors with a magnitude of one and are used to indicate direction. The standard unit vectors in two dimensions are i (1,0) and j (0,1). They serve as the building blocks for constructing other vectors, allowing for easy representation and manipulation of vector quantities in physics and engineering.