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Understanding the magnitude of a vector is crucial when dealing with vectors in mathematics and physics. The magnitude is essentially the "length" of the vector, a measure of how far it extends from the origin in a Cartesian coordinate system. In simple terms, it's how much ground a vector covers. The formula to find the magnitude of a vector \( \mathbf{a} = \langle x, y \rangle \) is given by:\[ \| \mathbf{a} \| = \sqrt{x^2 + y^2} \]This is derived from the Pythagorean Theorem, where a vector can be thought of as the hypotenuse of a right triangle. Let's apply this formula to the vector \( \mathbf{a} = \langle 0, 10 \rangle \):

• Set \( x = 0 \) and \( y = 10 \).
• Plug these values into the formula: \( \| \mathbf{a} \| = \sqrt{0^2 + 10^2} = \sqrt{100} \).
• Simplify to find \( \| \mathbf{a} \| = 10 \).

Thus, the magnitude of vector \( \mathbf{a} \) is 10. This tells us that the vector spans a distance of 10 units in whichever direction it's pointing.

The angle that a vector forms with the positive x-axis helps in understanding its orientation in a plane. This angle can tell us exactly how the vector is pointed relative to the standard reference direction—the positive x-axis. To find this angle \( \theta \), we typically use the arctangent function, which is defined as:\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]This formula calculates the angle based on the opposite and adjacent sides of a right triangle formed with the vector. For our vector \( \mathbf{a} = \langle 0, 10 \rangle \), since \( x = 0 \), the vector is purely vertical. In such cases:

• When \( y > 0 \), the vector points directly upwards, making \( \theta = 90^\circ \).

So in this case, the angle \( \theta \) is \( 90^\circ \), indicating that the vector is oriented straight up, perpendicular to the x-axis.

The direction of a vector tells us where exactly the vector is pointing, providing a clearer understanding of its orientation. This is represented by the angle it forms with some reference axis—in most cases, the positive x-axis.When describing the direction in a 2D plane, a vector like \( \mathbf{a} = \langle 0, 10 \rangle \) is straightforward because:

• It has no x-component, so it doesn't move left or right.
• All its "movement" is in the y-direction.

This specific direction is also linked with the previously calculated angle \( \theta = 90^\circ \). Since \( y = 10 \) and \( x = 0 \), the vector points exclusively upwards. So, we conclude that this vector's direction is vertically upward, aligned perfectly with the positive y-axis. Understanding the direction helps not only in visualizing the vector but also in applications where precise orientation matters.