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Vector magnitude is a measure that indicates the length or size of the vector. It is always a non-negative number. To find the magnitude of a vector denoted as \(\vec{A} = a \hat{\imath} + b \hat{\jmath}\), you use the Pythagorean theorem. This can be expressed as \(\|\vec{A}\| = \sqrt{a^2 + b^2}\).
For example, if you have a vector \(\vec{A} = 4.00 \hat{\imath} + 3.00 \hat{\jmath}\), the magnitude would be calculated by squaring each component, summing them, and then taking the square root:

• Calculate \((4.00)^2 = 16.00\)
• Calculate \((3.00)^2 = 9.00\)
• Add these results: \(16.00 + 9.00 = 25.00\)
• Find the square root: \(\sqrt{25.00} = 5.00\)

This process helps you determine how long the vector is when plotted graphically on a two-dimensional plane.

Vector subtraction is the operation of finding the difference between two vectors. It allows you to find the vector pointing from one vector's tip to the other's tip. To perform vector subtraction, you subtract corresponding components.
Consider two vectors, \(\vec{A} = 4.00 \hat{\imath} + 3.00 \hat{\jmath}\) and \(\vec{B} = 5.00 \hat{\imath} - 2.00 \hat{\jmath}\). The difference \(\vec{A} - \vec{B}\) is found by subtracting the components of \(\vec{B}\) from \(\vec{A}\):

• Subtract the \(\hat{\imath}\) components: \(4.00 - 5.00 = -1.00\)
• Subtract the \(\hat{\jmath}\) components: \(3.00 - (-2.00) = 5.00\)

Hence, the resulting vector \(\vec{A} - \vec{B}\) is \(-1.00 \hat{\imath} + 5.00 \hat{\jmath}\). This vector effectively describes the direction and distance from the tip of \(\vec{B}\) to the tip of \(\vec{A}\).

Vector components break down the vector along the coordinate axes, making it easier to apply various vector operations like addition, subtraction, or finding the magnitude. Each vector can be viewed as having two parts: the horizontal component along the \(\hat{\imath}\) axis and the vertical component along the \(\hat{\jmath}\) axis.
For example, the vector \(\vec{A} = 4.00 \hat{\imath} + 3.00 \hat{\jmath}\) has:

• A horizontal component: \(4.00 \hat{\imath}\)
• A vertical component: \(3.00 \hat{\jmath}\)

Understanding vector components is crucial as they allow you to calculate the vector’s magnitude and perform operations like vector addition or subtraction. Each component acts independently along its respective axis, which simplifies calculations involving vector relationships. By considering components, you can visualize how one vector can project onto others.