Question

Find the following vectors. The vector in the direction opposite that of (6,-8) with length 10

Short answer

Question: Determine the vector in the direction opposite that of (6, -8) with a length of 10. Answer: The vector is (-6, 8).

Step-by-step solution

Find the unit vector

First, we find the unit vector of the given vector (6, -8). The unit vector is the vector divided by its magnitude. The magnitude of a vector (x, y) can be calculated as: Magnitude = sqrt(x^2 + y^2) So, the magnitude of (6, -8) is: Magnitude = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10 Now, we can find the unit vector by dividing the given vector by its magnitude: Unit vector = (6/10, -8/10) = (3/5, -4/5)

Reverse the direction

To reverse the direction of the unit vector, we simply multiply it by -1: Reversed unit vector = -1 * (3/5, -4/5) = (-3/5, 4/5)

Multiply by the desired length

Finally, we multiply the reversed unit vector by the desired length, which is 10 in this case: New vector = 10 * (-3/5, 4/5) = (-6, 8) So the vector in the direction opposite that of (6, -8) with length 10 is (-6, 8).

Key concepts

Vector Magnitude

The concept of vector magnitude is essential in understanding the size or length of a vector. Think of it as measuring how much of the vector there is, similar to measuring the length of a line segment in geometry. In mathematical terms, for a vector represented as \((x, y)\), the magnitude is calculated using the Pythagorean theorem:\[\text{Magnitude} = \sqrt{x^2 + y^2}\]This formula gives the distance of the vector from the origin \((0, 0)\) to the point \((x, y)\) in a 2-dimensional plane, effectively capturing its size. For instance, the vector \((6, -8)\) has a magnitude of \(\sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = 10\).
This result helps us understand just how large or small a vector is, irrespective of the direction it points to. Knowing the magnitude is crucial when you need vectors of specific lengths, such as when scaling, normalizing, or finding specific vector directions.

Unit Vector

A unit vector is a vector that has a magnitude of exactly 1. Its primary role is to indicate the direction of a vector without regard to its magnitude. This makes the unit vector ideal for finding direction, while keeping its size constant. To derive a unit vector from any given vector, you simply divide each component of the vector by the vector's magnitude. For the vector \((6, -8)\), which has a magnitude of 10, the unit vector becomes:

• \((6/10, -8/10)\) or simplified, \((3/5, -4/5)\)

This conversion results in a vector that points in the same direction as \((6, -8)\) but measures precisely 1 unit long. Using unit vectors is particularly useful in applications that involve direction, as they simplify interactions between vectors by standardizing their size.

Vector Direction

The direction of a vector is a fundamental aspect as it determines where the vector points in space. The direction can be altered by reversing or rotating the vector, while still retaining its original magnitude. In your exercise, you reversed the direction of a unit vector \((3/5, -4/5)\) by multiplying by -1 to get \((-3/5, 4/5)\).

This opposite direction creates a mirror effect, flipping the vector's orientation. When considering directions, it's possible to move towards or away from something, and altering vector direction helps achieve this. Finally, manipulating direction often involves scaling the unit vector to the desired length, ensuring that the vector points correctly while being the required size. For example, multiplying \((-3/5, 4/5)\) by 10 gives \((-6, 8)\), a vector of length 10 in the opposite direction to \((6, -8)\). Understanding and adjusting vector direction is crucial in scenarios such as navigation, physics calculations, and any field that requires precise path control.