What is the difference between a vector's magnitude and its direction?

Magnitude is a scalar value representing the size or length of the vector, calculated via Pythagoras. Direction is the orientation of the vector, usually expressed as an angle relative to a coordinate axis.

How does a unit vector differ from the original vector it was derived from?

A unit vector has the exact same direction as the original vector but its magnitude is scaled to exactly 1. It is found by dividing the original vector by its magnitude.

When calculating the direction of a vector, why is a sketch often necessary?

The inverse tangent function only returns values between $-90^{\circ}$ and $90^{\circ}$. A sketch helps identify which quadrant the vector is in so you can adjust the angle to the correct range (e.g., adding $180^{\circ}$ for the third quadrant).

What is a common error when calculating the magnitude of a vector like $-3\mathbf{i} + 4\mathbf{j}$?

A common mistake is failing to square the negative sign correctly, resulting in $-9 + 16 = 7$. Squaring any component must result in a positive value ($(-3)^2 = 9$), so the sum should be $9 + 16 = 25$.

What happens to the direction of a vector if it is multiplied by a negative scalar?

Multiplying by a negative scalar reverses the direction of the vector (a $180^{\circ}$ change). The magnitude is also scaled by the absolute value of that scalar.

Why can you not simply use $\theta = \tan^{-1}(y/x)$ for all vectors?

This formula can lead to errors if $x=0$ (division by zero) or if the vector is in the 2nd or 3rd quadrants where the x-coordinate is negative, as the calculator cannot distinguish between $\frac{y}{-x}$ and $\frac{-y}{x}$.

Define the 'modulus' of a vector.

The modulus is another term for the magnitude or size of the vector. It is the scalar length of the arrow representing the vector in space.

What is the formula for the magnitude of vector $\mathbf{v} = x\mathbf{i} + y\mathbf{j}$?

The magnitude is given by $|\mathbf{v}| = \sqrt{x^2 + y^2}$. This is derived from the Pythagorean theorem where the components are the legs of a right triangle.

How do you calculate the x and y components if you are given magnitude $r$ and angle $\theta$?

The components are found using $x = r \cos \theta$ and $y = r \sin \theta$. This process is known as resolving the vector into its horizontal and vertical parts.

What is the magnitude of any unit vector?

By definition, the magnitude of any unit vector is exactly 1. This makes it a 'pure' direction with no scaling effect when used in multiplications.

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Magnitude & Direction

Summary

Magnitude and direction are the two fundamental properties that define a vector. Magnitude represents the scalar size or length of the vector, while direction specifies its orientation in space, typically measured as an angle relative to a fixed axis. Understanding how to calculate these properties and convert between component form and polar form is essential for vector analysis in physics and engineering.

1. Definition & Core Concepts

Magnitude refers to the length or 'size' of a vector. In a geometric sense, it represents the straight-line distance from the starting point (tail) to the end point (head) of the vector.
The magnitude of a vector $\mathbf{a}$ is denoted by $|\mathbf{a}|$ or sometimes referred to as its modulus.
Direction is the orientation of the vector in a 2D plane. By convention, it is usually measured as an angle $\theta$ anticlockwise from the positive x-axis.
A vector is fully defined only when both its magnitude and direction are known; changing either one results in a different vector.

A vector diagram showing the magnitude as the hypotenuse of a right-angled triangle formed by x and y components, with the angle theta measured from the x-axis.

2. Calculating Magnitude

3. Determining Direction

4. Unit Vectors

5. Key Distinctions

6. Exam Strategy & Tips