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Chapter 10: Problem 6

What effect does multiplying a vector by -2 have?

Short Answer

Expert verified
Multiplying a vector by -2 doubles its length and reverses its direction.

Step by step solution

01

Understand Vector Multiplication by Scalar

When a vector in space is multiplied by a scalar (a real number), each component of the vector is multiplied by that scalar. If you have a vector \( \mathbf{v} = (v_1, v_2, v_3) \), multiplying it by a scalar \( k \) gives \( k\mathbf{v} = (kv_1, kv_2, kv_3) \).
02

Consider Multiplying by -2

For the scalar \( -2 \), multiplying a vector \( \mathbf{v} = (v_1, v_2, v_3) \) by \(-2\) results in \(-2\mathbf{v} = (-2v_1, -2v_2, -2v_3)\). Each component of the vector is scaled by 2 and also changes sign (positive to negative or negative to positive).
03

Interpret the Geometric Effect

Multiplying a vector by \(-2\) effectively scales the vector by a factor of 2, which means the magnitude or length of the vector is doubled. Additionally, multiplying by a negative number reverses its direction, resulting in a vector extending in the opposite direction from the original.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
When dealing with vectors, scalar multiplication is a fundamental operation. It involves multiplying a vector by a scalar, which is simply a real number. This operation affects each component of the vector.
For example, if you have a vector \( \mathbf{v} = (v_1, v_2, v_3) \), and you multiply it by a scalar \( k \), the result is \( k\mathbf{v} = (kv_1, kv_2, kv_3) \).
  • Each component of the vector is multiplied by the scalar.
  • Scalar multiplication can change the length and direction of the vector.
  • If the scalar is negative, the direction of the vector reverses.
Understanding scalar multiplication is crucial to grasping how vectors change under different conditions. It can help in visualizing both physics and geometry problems where vectors are involved.
Geometric Interpretation
The geometric interpretation of scalar multiplication can be quite illuminating. When a vector is multiplied by a scalar, this operation can change its scale and orientation.
If you multiply a vector by a positive number, it will grow longer if the scalar has an absolute value greater than 1. Conversely, it will shrink if the scalar is between 0 and 1.
With our specific problem, multiplying by \(-2\), the negative scalar means:
  • The vector's magnitude is doubled.
  • Its direction is reversed.
This results in a vector that is not only twice the length of the original but also pointing in the opposite direction. This double transformation by a negative scalar is a powerful tool in vector mathematics.
Vector Direction
The direction of a vector is a vital attribute as it signifies where the vector is pointing. When multiplying by negative scalars, it results in a direction change.
When you multiply a vector \( \mathbf{v} \) by \(-1\), the vector flips direction. If absorbed into a larger scalar like \(-2\), not only is the direction inverted, but the magnitude is altered as well.
  • Direction is reversed with a negative scalar.
  • Important in applications involving motion or forces.
Rather than adding complex motion dynamics, scalar multiplication provides a simple mathematical operation to visualize these changes directly.
Vector Magnitude
Magnitude refers to the length of a vector. In practical terms, it tells us how much of the vector there is, regardless of its direction.
When a vector \( \mathbf{v} \) is multiplied by a scalar, its magnitude is scaled by the absolute value of the scalar. So, for a scalar \(-2\):
  • The magnitude is multiplied by \(2\). This is because magnitude, being length, is always positive and responds to scalar multiplication in terms of absolute value.
  • Magnitude change is independent of direction change.
Hence, multiplying by \(-2\) results in doubling the magnitude, making vectors longer, but also affecting direction if the scalar is negative. This understanding helps facilitate computations and visualizations in vector-based calculations.

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Most popular questions from this chapter

Give the equation of the described plane in standard and general forms. Contains the point (2,-6,1) and the line $$ \ell(t)=\left\\{\begin{array}{l} x=2+5 t \\ y=2+2 t \\ z=-1+2 t \end{array}\right. $$

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(-2,5)\), parallel to \(\vec{d}=\langle 0,1\rangle\).

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(0,1,2)\) and orthogonal to both \(\vec{d}_{1}=\langle 2,-1,7\rangle\) and \(\vec{d}_{2}=\langle 7,1,3\rangle\)

Give any two points in the given plane. \(4(y+2)-(z-6)=0\)

Give the equation of the described plane in standard and general forms. Contains the point (1,2,3) and is parallel to the plane \(x=5 .\)
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