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Chapter 13: Problem 29

Find a unit vector in the opposite direction to \(2 \vec{i}-\vec{j}-\) \(\sqrt{11} \vec{k}\)

Short Answer

Expert verified
The unit vector is \(-\frac{1}{2}\vec{i} + \frac{1}{4}\vec{j} + \frac{\sqrt{11}}{4}\vec{k}\)."

Step by step solution

01

Write the Given Vector

The vector given is \(\mathbf{v} = 2\vec{i} - \vec{j} - \sqrt{11}\vec{k}\). We need to find a unit vector in the opposite direction.
02

Find the Opposite Vector

To find the opposite vector, we simply negate each component of the vector \( \mathbf{v} \). Thus, the opposite vector is \(-\mathbf{v} = -2\vec{i} + \vec{j} + \sqrt{11}\vec{k}\).
03

Calculate the Magnitude of the Opposite Vector

The magnitude of the vector \(-\mathbf{v}\) is given by \( \| -\mathbf{v} \| = \sqrt{(-2)^2 + 1^2 + (\sqrt{11})^2} \). Simplify this as \( \sqrt{4 + 1 + 11} = \sqrt{16} = 4 \).
04

Find the Unit Vector in the Opposite Direction

The unit vector \(\mathbf{u}\) in the opposite direction is obtained by dividing each component of \(-\mathbf{v}\) by its magnitude. Thus, \(\mathbf{u} = \frac{-2\vec{i} + \vec{j} + \sqrt{11}\vec{k}}{4} = -\frac{1}{2}\vec{i} + \frac{1}{4}\vec{j} + \frac{\sqrt{11}}{4}\vec{k}\).

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Key Concepts

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

Vector Negation
When dealing with vectors, the concept of vector negation is essential for finding vectors that point in the opposite direction. Negating a vector means reversing the direction of all its components. For any vector \(\mathbf{v} = a\vec{i} + b\vec{j} + c\vec{k}\), its negated vector, \(-\mathbf{v} \), is obtained by simply changing the signs of each component. Thus, we have:
  • -\(a\) becomes \(-a\)
  • -\(b\) becomes \(-b\)
  • -\(c\) becomes \(-c\)
Negating a vector does not change its magnitude, only its direction. This process is straightforward: if you know the original vector, just multiply each component by -1 to find its opposite.
Vector Magnitude
Understanding the magnitude of a vector is crucial for many vector calculations, such as normalizing a vector to create a unit vector. The magnitude of a vector \(\mathbf{v} = a\vec{i} + b\vec{j} + c\vec{k}\) can be found using the formula: \[\| \mathbf{v} \| = \sqrt{a^2 + b^2 + c^2}\]This formula derives from the Pythagorean theorem, extended into three dimensions. Calculating the magnitude involves squaring each component, summing these squares, and taking the square root of the result.

The magnitude gives a sense of the 鈥渓ength鈥� or 鈥渟ize鈥� of the vector in space. For example, in the solution provided, the magnitude helps adjust the negated vector to become a unit vector.
Direction of Vectors
Vectors are unique because, unlike scalar quantities, they have a direction. Expressing a vector's direction is key when it needs to be normalized or transformed into a unit vector.

The direction of any vector \(\mathbf{v}\) can be understood as the way in which its components point in space. You can visualize the direction by observing how much influence each basis vector (like \(\vec{i}, \vec{j}, \vec{k}\)) has on the overall vector.
  • An important aspect of vector direction is when we create a unit vector. The unit vector maintains the original direction of \(\mathbf{v}\) but shrinks its magnitude to 1. We do this by dividing each component by the vector's magnitude as shown in the exercise.
  • The outcome is a vector that points in the same direction as the negated vector but on a normalized scale, ensuring it has a length of exactly 1.
This concept is particularly useful in applications where precise direction is more critical than the magnitude, such as in orientation calculations in physics and engineering.

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Most popular questions from this chapter

Find an equation of a plane that satisfies the given conditions. Parallel to \(2 x+4 y-3 z=1\) and through (1,0,-1)

Give reasons for your answer. If \(\vec{u} \cdot \vec{v}=\|\vec{u}\|\|\vec{v}\|\) then \(\|\vec{u}+\vec{v}\|=\|\vec{u}\|+\|\vec{v}\|\)

(a) Find a vector perpendicular to the plane \(z=2+3 x-y\) (b) Find a vector parallel to the plane.

Give an example of: A vector \(\vec{v}\) such that \(\|\vec{u} \times \vec{v}\|=10,\) where \(\vec{u}=3 \vec{i}+4 \vec{j}\)

Are the statements true or false? Give reasons for your answer. \(\vec{u} \times \vec{v}\) is a vector.
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