In Exercises 22–24, sketch each vector as a position vector and find its magnitude. v = -3i - 4j

Verified step by step guidance

1. Step 1: Understand that the vector \( \mathbf{v} = -3\mathbf{i} - 4\mathbf{j} \) is given in component form, where \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors along the x-axis and y-axis, respectively.
2. Step 2: To sketch the vector as a position vector, plot the point \((-3, -4)\) on the Cartesian coordinate system. The vector \( \mathbf{v} \) is represented as an arrow starting from the origin \((0, 0)\) and pointing to the point \((-3, -4)\).
3. Step 3: To find the magnitude of the vector \( \mathbf{v} \), use the formula for the magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), which is \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \).
4. Step 4: Substitute the components of the vector into the magnitude formula: \( a = -3 \) and \( b = -4 \).
5. Step 5: Calculate the expression \( \sqrt{(-3)^2 + (-4)^2} \) 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 two-dimensional Cartesian coordinate system, it is expressed in terms of its components along the x-axis and y-axis, typically denoted as v = xi + yj, where x and y are the coordinates. For the vector v = -3i - 4j, the position vector indicates a point located at (-3, -4) in the plane.

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 the formula |v| = √(x² + y²). In the case of v = -3i - 4j, the magnitude would be |v| = √((-3)² + (-4)²) = √(9 + 16) = √25 = 5.

Vector Components

Vector components break down a vector into its individual parts along the coordinate axes. In the vector v = -3i - 4j, the component -3 corresponds to the x-direction (horizontal) and -4 to the y-direction (vertical). Understanding these components is essential for visualizing the vector's direction and calculating its magnitude.