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The magnitude of a vector is a measure of its length or size, often represented in geometrical terms. Imagine this as if you were measuring the distance a straight line would travel from one point to another. In mathematical terms, if we have a vector \( \mathbf{v} = \langle a, b \rangle \), its magnitude \( \| \mathbf{v} \| \) is calculated using the formula:

• \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \)

This formula is derived from the Pythagorean theorem. It tells us that to find the magnitude, or length, of the vector, we take the square root of the sum of the squares of its components.

When solving problems like the one in the exercise, setting up the correct magnitude equation is crucial. In the given problem, the vector \( \mathbf{v} \) was supposed to have a magnitude of \( \sqrt{5} \). Thus, using the given vector components \( \langle -1, y_2 \rangle \), we set up the equation \( \sqrt{(-1)^2 + y_2^2} = \sqrt{5} \) to ensure our vector \( \mathbf{v} \) reaches the desired length.

Coordinates are numerical values that locate a point in space, like a map's address for a place. For a two-dimensional space, coordinates are written in pairs, often \((x, y)\), representing horizontal and vertical positions on a plane.

In the exercise, the initial point \( P \) is given coordinates \((1,0)\), with \( 1 \) marking the position along the x-axis (horizontal axis) and \( 0 \) on the y-axis (vertical axis). The terminal point \( Q \), which we needed to find, is specifically on the y-axis—indicating its x-coordinate is \( 0 \).

This characteristic of \( Q \) being on the y-axis was key as it directly informed us how to set the x-values when defining the vector \( \mathbf{v} \). It simplifies calculations, showing how coordinates provide essential data to structure our vector problem-solving strategies.

Vector components are the building blocks of a vector, breaking down its direction and magnitude into manageable parts. If you picture a vector as an arrow, its components \( \langle a, b \rangle \) show how the arrow extends in two perpendicular directions often aligned with the axes.

In the problem, the vector \( \mathbf{v} \) has components derived from subtracting the coordinates of its initial point from its terminal point. Mathematically, if point \( P(1,0) \) is the initial and \( Q \) is the terminal, the vector is represented as:

• \( \mathbf{v} = \langle x_2 - 1, y_2 - 0 \rangle \).

Knowing that \( Q \) lies on the y-axis simplifies this to \( \langle -1, y_2 \rangle \), indicating the vector moves one unit left on the x-axis and upwards by \( y_2 \) units.

Comprehending how vector components work not only helps in setting up the vector but also assists in determining other vector properties like magnitude.