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

Verified step by step guidance

1. Calculate the magnitude of vector E, which is represented as $E=2i+3j$. Use the formula for the magnitude of a vector: $|E|=\sqrt{E_x^2+E_y^2}$, where $E_x$ and $E_y$ are the components of vector E along the x and y axes, respectively.
2. Substitute the components of vector E into the magnitude formula: $|E|=\sqrt{2^2+3^2}$.
3. Calculate the magnitude of vector F, which is represented as $F=2i-2j$. Use the same formula for the magnitude of a vector: $|F|=\sqrt{F_x^2+F_y^2}$, where $F_x$ and $F_y$ are the components of vector F along the x and y axes, respectively.
4. Substitute the components of vector F into the magnitude formula: $|F|=\sqrt{2^2+(-2{)}^2}$.
5. Simplify the expressions obtained in steps 2 and 4 to find the magnitudes of vectors E and F.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length or size, calculated using the Pythagorean theorem. For a vector represented as E = ai + bj, the magnitude is given by |E| = √(a² + b²). This concept is essential for determining how 'strong' or 'large' a vector is in a given space.

Vector Components

Vectors can be broken down into components along the coordinate axes, typically represented as i (horizontal) and j (vertical) in two-dimensional space. For example, the vector E = 2i + 3j has components 2 and 3, which indicate its influence in the x and y directions, respectively. Understanding components is crucial for vector addition and magnitude calculations.

Vector Addition

Vector addition involves combining two or more vectors to form a resultant vector. This is done by adding their corresponding components: if E = ai + bj and F = ci + dj, then E + F = (a+c)i + (b+d)j. This concept is fundamental when analyzing multiple forces or movements acting simultaneously in physics.