In Exercises 5–12, sketch each vector as a position vector and find its magnitude.v = -6i - 2j

Verified step by step guidance

1. Step 1: Understand the vector notation. The vector \( \mathbf{v} = -6\mathbf{i} - 2\mathbf{j} \) is given in terms of its components along the x-axis and y-axis. Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the direction of the x-axis and y-axis, respectively.
2. Step 2: Sketch the vector. To sketch the vector \( \mathbf{v} = -6\mathbf{i} - 2\mathbf{j} \), start at the origin (0,0) on a coordinate plane. Move 6 units to the left along the x-axis (since the x-component is -6) and 2 units down along the y-axis (since the y-component is -2). Draw an arrow from the origin to this point (-6, -2).
3. Step 3: Use the Pythagorean theorem to find the magnitude of the 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 formula. Here, \( a = -6 \) and \( b = -2 \). So, the magnitude is \( \|\mathbf{v}\| = \sqrt{(-6)^2 + (-2)^2} \).
5. Step 5: Simplify the expression under the square root. Calculate \( (-6)^2 \) and \( (-2)^2 \), then add the results to find the magnitude.

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 = -6i - 2j, the position vector indicates a point located 6 units left and 2 units down from the origin.

Vector Magnitude

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 this case, for v = -6i - 2j, the magnitude would be |v| = √((-6)² + (-2)²) = √(36 + 4) = √40, which simplifies to 2√10.

Vector Components

Vector components are the projections of a vector along the coordinate axes. In a two-dimensional space, a vector can be broken down into its horizontal (i) and vertical (j) components. For the vector v = -6i - 2j, the component -6 represents movement along the x-axis, while -2 represents movement along the y-axis, indicating the direction and distance in each dimension.