Question

Triple Choice Suppose that each component of a vector is doubled. (a) Does the magnitude of the vector increase, decrease, or stay the same? Explain. (b) Does the direction angle of the vector increase, decrease, or stay the same?

Short answer

(a) The magnitude increases; (b) The direction stays the same.

Step-by-step solution

Understanding Vector Magnitude

The magnitude of a vector \( \mathbf{v} = \langle x, y \rangle \) is calculated using the formula \( |\mathbf{v}| = \sqrt{x^2 + y^2} \). If each component of the vector is doubled, the new vector \( \mathbf{v'} = \langle 2x, 2y \rangle \) would have a magnitude given by \( |\mathbf{v'}| = \sqrt{(2x)^2 + (2y)^2} \). Simplifying, this results in \( |\mathbf{v'}| = \sqrt{4x^2 + 4y^2} = \sqrt{4(x^2 + y^2)} = 2\sqrt{x^2 + y^2} = 2|\mathbf{v}| \). Thus, the magnitude of the vector doubles.

Impact on Magnitude

Since doubling the components of a vector results in doubling the magnitude, the magnitude of the vector increases.

Understanding Vector Direction

The direction of a vector is related to its angle with respect to the x-axis, calculated as \( \theta = \tan^{-1}(\frac{y}{x}) \). When both components of a vector are doubled, \( \mathbf{v'} = \langle 2x, 2y \rangle \), the direction angle is \( \theta' = \tan^{-1}(\frac{2y}{2x}) = \tan^{-1}(\frac{y}{x}) \). This is identical to the original vector’s direction angle, indicating that the direction remains unchanged.

Impact on Direction

Doubling the vector's components does not change the ratio \( \frac{y}{x} \), hence, the direction angle of the vector remains the same.

Key concepts

Vector Components

A vector, in two-dimensional space, can be described by its components along the x and y axes. These components can be seen as the horizontal and vertical projections of the vector. For example, for the vector \( \mathbf{v} = \langle x, y \rangle \), the values \( x \) and \( y \) are its components. When you talk about changing the components of a vector, you are essentially adjusting its length or direction—or both—depending on how you adjust those components. For example, if each component, \( x \) and \( y \) of a vector is multiplied by the same scalar factor, say 2, then the vector doubles in size. But what about its direction? We'll get to that!Remember:

• Components are treated separately, as they relate to dimensions in a coordinate system.
• The magnitude and direction depend on these components.

Understanding vector components thoroughly is foundational for further operations like scaling and calculations of magnitude and direction.

Scalar Multiplication

Scalar multiplication is a process where a vector is multiplied by a scalar, which is just a fancy term for a regular number. This operation alters the magnitude of the vector but not its direction, provided the scalar is positive. For example, in our case, if the vector \( \mathbf{v} = \langle x, y \rangle \) is multiplied by a scalar 2, the resulting vector \( \mathbf{v'} = \langle 2x, 2y \rangle \). The components are doubled, leading to changes in its length:

• New x-component: \( 2x \)
• New y-component: \( 2y \)

Although the magnitude alters, the vector maintains its direction because the components have been scaled uniformly. When performing scalar multiplication:

• The vector stretches (for a factor > 1) or shrinks (if between 0 and 1).
• The direction remains constant (if the scalar is positive).

This concept is crucial for understanding how scaling a vector affects only its size.

Direction Angle

The direction of a vector refers to the angle it makes with the positive x-axis. This is an important property because it helps indicate towards which quadrant or direction the vector is pointing.The angle \( \theta \) for a vector \( \mathbf{v} = \langle x, y \rangle \) is calculated using the formula \( \theta = \tan^{-1}(\frac{y}{x}) \). This trigonometric function gives the angle based on the relationship between the vertical and horizontal components.Interestingly, if you double both the x and y components of the vector to get \( \mathbf{v'} = \langle 2x, 2y \rangle \), the angle doesn't change. That's because the ratio \( \frac{y}{x} \) remains unchanged, and so \( \theta' = \theta \). Therefore, the vector stretches along its line without altering its orientation.Key notes:

• The ratio of the components determines the direction angle.
• As long as the ratio remains constant, the direction stays the same.

Understanding the direction angle is essential for describing a vector's orientation precisely.

Vector Calculation

Vectors require careful calculations when determining properties like magnitude and direction. The vector \( \mathbf{v} = \langle x, y \rangle \) has a magnitude given by the formula \( |\mathbf{v}| = \sqrt{x^2 + y^2} \). This measures the length or size of the vector.Consider when each component of the vector is doubled, leading to \( \mathbf{v'} = \langle 2x, 2y \rangle \). The magnitude for this new vector is calculated as:\[ |\mathbf{v'}| = \sqrt{(2x)^2 + (2y)^2} = \sqrt{4x^2 + 4y^2} = 2\sqrt{x^2 + y^2} = 2|\mathbf{v}| \]Thus, the doubling of the components results in a doubling of the magnitude.For the direction, as said before: \[ \theta' = \tan^{-1}\left(\frac{2y}{2x}\right) = \tan^{-1}\left(\frac{y}{x}\right) = \theta \]Meaning the angle—and therefore the direction—stays the same.Summing up vector calculations:

• The magnitude changes with scalar multiplication.
• The direction, often given by an angle, remains constant if the vector is scaled uniformly by a positive scalar.

Knowing how to perform these calculations helps fully describe a vector's transformation when its components are modified.