In Exercises 5–12, sketch each vector as a position vector and find its magnitude.v = -4i

Verified step by step guidance

1. Step 1: Understand that the vector \( \mathbf{v} = -4\mathbf{i} \) is a position vector in the Cartesian coordinate system, where \( \mathbf{i} \) is the unit vector along the x-axis.
2. Step 2: Sketch the vector \( \mathbf{v} = -4\mathbf{i} \) on a coordinate plane. Start at the origin (0,0) and move 4 units to the left along the x-axis, since the vector is negative.
3. Step 3: To find the magnitude of the vector \( \mathbf{v} = -4\mathbf{i} \), use the formula for the magnitude of a vector \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the components of the vector.
4. Step 4: Substitute the components of the vector into the magnitude formula. Here, \( a = -4 \) and \( b = 0 \), so the magnitude is \( \|\mathbf{v}\| = \sqrt{(-4)^2 + 0^2} \).
5. Step 5: Simplify the expression under the square root to find the magnitude of the vector.

Key Concepts

Position Vectors

A position vector represents a point in space relative to an origin. In a Cartesian coordinate system, it is expressed in terms of its components along the axes, such as 'v = -4i', which indicates a vector pointing 4 units in the negative x-direction. Understanding position vectors is essential for visualizing and manipulating vectors in two or three dimensions.

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 expressed in component form, like 'v = -4i', the magnitude is found by taking the square root of the sum of the squares of its components. In this case, the magnitude is |v| = √((-4)²) = 4, indicating the distance from the origin to the point represented by the vector.

Unit Vectors

A unit vector is a vector with a magnitude of one, used to indicate direction without regard to length. It is often derived from a given vector by dividing the vector by its magnitude. Understanding unit vectors is important for normalizing vectors and for applications in physics and engineering, where direction is crucial.