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

Verified step by step guidance

1. <Step 1: Understand the vector notation. The vector \( \mathbf{v} = -3\mathbf{j} \) is given in terms of the unit vector \( \mathbf{j} \), which represents the direction along the y-axis.>
2. <Step 2: Sketch the vector. Since \( \mathbf{v} = -3\mathbf{j} \), this means the vector points in the negative y-direction with a magnitude of 3. On a coordinate plane, start at the origin (0,0) and draw a line 3 units downwards along the y-axis.>
3. <Step 3: Recall 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: Apply the formula to find the magnitude. In this case, \( a = 0 \) and \( b = -3 \). Substitute these values into the formula: \( \|\mathbf{v}\| = \sqrt{0^2 + (-3)^2} \).>
5. <Step 5: Simplify the expression under the square root. Calculate \( (-3)^2 \) and then take 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 two-dimensional Cartesian coordinate system, a position vector is expressed in terms of its components along the x and y axes. For example, the vector v = -3j indicates a point that is 3 units in the negative y-direction, with no displacement along the x-axis.

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 as v = ai + bj, the magnitude is given by the formula |v| = √(a² + b²). In the case of v = -3j, the magnitude is simply the absolute value of the y-component, which is 3.

Unit Vectors

A unit vector is a vector that has a magnitude of one and indicates direction. It is often used to represent the direction of a vector without regard to its magnitude. To convert a vector into a unit vector, you divide the vector by its magnitude. For the vector v = -3j, the unit vector would be -j, indicating the same direction but with a magnitude of 1.