Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, -7〉

Verified step by step guidance

1. Step 1: To find the magnitude of the vector \( \langle -4, -7 \rangle \), use the formula for the magnitude of a vector \( \langle a, b \rangle \), which is \( \sqrt{a^2 + b^2} \). Substitute \( a = -4 \) and \( b = -7 \) into the formula.
2. Step 2: Calculate the magnitude by evaluating \( \sqrt{(-4)^2 + (-7)^2} \). This involves squaring each component, adding the results, and then taking the square root.
3. Step 3: To find the direction angle \( \theta \) of the vector, use the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Substitute \( a = -4 \) and \( b = -7 \) into the formula.
4. Step 4: Calculate \( \theta = \tan^{-1}\left(\frac{-7}{-4}\right) \). This will give you the angle in radians or degrees, depending on your calculator settings.
5. Step 5: Since the vector \( \langle -4, -7 \rangle \) is in the third quadrant (both components are negative), adjust the angle \( \theta \) by adding 180 degrees to ensure it reflects the correct direction in standard position.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector represented as 〈x, y〉, the magnitude is given by the formula √(x² + y²). In this case, for the vector 〈-4, -7〉, the magnitude would be √((-4)² + (-7)²) = √(16 + 49) = √65.

Direction Angle

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the arctangent function: θ = arctan(y/x). For the vector 〈-4, -7〉, the angle can be calculated as θ = arctan(-7/-4), which will yield an angle in the third quadrant since both components are negative.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x and y coordinates. The first quadrant has both coordinates positive, the second has a negative x and positive y, the third has both negative, and the fourth has a positive x and negative y. Understanding the quadrant is essential for determining the correct angle measure, as angles in different quadrants require adjustments to ensure they are expressed correctly.