Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of (c) -E - 2F

Verified step by step guidance

1. First, calculate the vector -E by multiplying each component of vector E by -1. This gives -E = -2i - 3j.
2. Next, calculate the vector 2F by multiplying each component of vector F by 2. This results in 2F = 4i - 4j.
3. Now, add the vectors -E and 2F together. Add the corresponding components: (-2i - 3j) + (4i - 4j) = (4i - 2i) + (-4j - 3j) = 2i - 7j.
4. To find the magnitude of the resulting vector (2i - 7j), use the magnitude formula for vectors: $\text{Magnitude}=\sqrt{i^2+j^2}$.
5. Substitute the i and j components from the vector (2i - 7j) into the magnitude formula and calculate: $\text{Magnitude}=\sqrt{(2{)}^2+(-7{)}^2}$.

Key Concepts

Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. For example, if vector E has components (2, 3) and vector F has components (2, -2), then -E would be (-2, -3) and -2F would be (-4, 4). The resultant vector is found by summing these component-wise.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a 2D vector with components (x, y). This value represents the distance from the origin to the point defined by the vector in the Cartesian plane. For example, the magnitude of vector (a, b) is √(a² + b²).

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, multiplying vector F = (2, -2) by -2 results in the vector (-4, 4). This operation is crucial for adjusting the size of vectors in calculations.