Question

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 angle stays the same.

Step-by-step solution

Understanding Vector Magnitude

The magnitude of a vector $\mathbf{v}=(x,y)$ is calculated using the formula $||\mathbf{v}||=\sqrt{x^2+y^2}$. This represents the length of the vector.

Doubling the Vector Components

Doubling each component of $\mathbf{v}=(x,y)$ results in a new vector ${\mathbf{v}}^{\prime }=(2x,2y)$.

Calculating New Magnitude

The magnitude of the new vector ${\mathbf{v}}^{\prime }=(2x,2y)$ is $||{\mathbf{v}}^{\prime }||=\sqrt{(2x{)}^2+(2y{)}^2}=\sqrt{4x^2+4y^2}=\sqrt{4(x^2+y^2)}=2\sqrt{x^2+y^2}=2||\mathbf{v}||$. Thus, the magnitude doubles.

Evaluating Direction Angle

The direction angle of a vector is given by $\theta =\arctan \left(\frac{y}{x}\right)$. When both components of the vector are doubled, the ratio $\frac{2y}{2x}=\frac{y}{x}$ remains unchanged, meaning the direction angle $\theta$ stays the same.

Conclusion

Therefore, doubling the components of a vector results in doubling its magnitude, but the direction angle remains unchanged.

Key concepts

Vector Magnitude

Understanding the magnitude of a vector is crucial in vector mathematics. The magnitude represents the vector's length or how "long" it is in a geometric sense. For a vector $\mathbf{v}=(x,y)$, the magnitude $||\mathbf{v}||$ is calculated using the formula:$$||\mathbf{v}||=\sqrt{x^2+y^2}$$This formula is derived from the Pythagorean theorem, considering that a vector can be visualized as a line segment forming a right triangle with its components.

To see how changes in the vector affect its magnitude, consider doubling each component. If ${\mathbf{v}}^{\prime }=(2x,2y)$, then $||{\mathbf{v}}^{\prime }||=2||\mathbf{v}||$. This shows that the magnitude scales directly with any scaling of the components.

Vector Components

Vector components are the individual parts that make up the vector. For a two-dimensional vector like $\mathbf{v}=(x,y)$, 'x' and 'y' are its components. These components can be thought of as projections of the vector in a coordinate system.

• The 'x' component affects the horizontal movement.
• The 'y' component affects the vertical movement.

Doubling these components, as in our exercise where ${\mathbf{v}}^{\prime }=(2x,2y)$, means each part of the vector stretches by that factor. Component scaling impacts how vectors interact with forces, velocities, and within transformations.

Direction Angle

The direction angle of a vector is a crucial concept describing its orientation. For vector $\mathbf{v}=(x,y)$, the direction angle $\theta$ is calculated by:$$\theta =\arctan \left(\frac{y}{x}\right)$$This calculation uses the tangent ratio from trigonometry, determining the vector's angle relative to the positive x-axis.

When both components of a vector are doubled, such as ${\mathbf{v}}^{\prime }=(2x,2y)$, the ratio $\frac{2y}{2x}$ is equivalent to $\frac{y}{x}$. Hence, the direction angle $\theta$ remains unchanged, indicating that doubling shifts the vector further along the same path.

Vector Scaling

Vector scaling involves multiplying all components of a vector by a constant. This operation affects the vector's magnitude but not its direction angle. For instance, multiplying $\mathbf{v}=(x,y)$ by a factor of 2 results in ${\mathbf{v}}^{\prime }=(2x,2y)$.

• Scaling the components directly increases or decreases the vector's size.
• The vector's direction remains constant since the angle with the x-axis doesn’t change.

This property of maintaining the direction while adjusting magnitude makes vector scaling particularly useful in physics and engineering for modeling real-world phenomena like forces or velocity changes.