Problem 24 - APEX Calculus

Question

Find the unit vector \(\vec{u}\) in the direction of \(\vec{v} .\) \(\vec{v}=\langle 1,-2,2\rangle\)

Step-by-Step Solution

Answer

The unit vector \(\vec{u}\) is \(\left\langle \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right\rangle\).

Step 1: Understanding the Problem

We are given a vector \(\vec{v} = \langle 1, -2, 2 \rangle\) and need to find a unit vector \(\vec{u}\) in the direction of \(\vec{v}\). A unit vector has a magnitude (length) of 1 and points in the direction of the given vector.

Step 2: Compute the Magnitude of \(\vec{v}\)

To find the magnitude of \(\vec{v}\), use the formula for the magnitude of a vector: \[||\vec{v}|| = \sqrt{1^2 + (-2)^2 + 2^2} \]Calculate:\[||\vec{v}|| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\]

Step 3: Calculate the Unit Vector \(\vec{u}\)

The unit vector \(\vec{u}\) in the direction of \(\vec{v}\) is given by scaling \(\vec{v}\) by the reciprocal of its magnitude. So, \[\vec{u} = \frac{1}{||\vec{v}||} \cdot \vec{v} = \frac{1}{3} \cdot \langle 1, -2, 2 \rangle\]Calculate the components:\[\vec{u} = \left\langle \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right\rangle\]

Key Concepts

Vector Magnitude

The concept of vector magnitude is essential in identifying not just how a vector directs, but how much it extends in space. Essentially, magnitude determines the size or length of a vector. When working with any vector, say \( \vec{v} = \langle a, b, c \rangle \), the magnitude is calculated using the Pythagorean formula:\[ ||\vec{v}|| = \sqrt{a^2 + b^2 + c^2} \]For example, consider the vector \( \vec{v} = \langle 1, -2, 2 \rangle \). Calculating its magnitude involves squaring each component, adding those squares, and taking the square root of the total.

• Square each component: \( 1^2 = 1 \), \((-2)^2 = 4 \), \(2^2 = 4 \)
• Add them up: \(1 + 4 + 4 = 9 \)
• Take the square root: \(\sqrt{9} = 3 \)

Hence, the vector \( \vec{v} \) has a magnitude of 3. The magnitude helps us quantify how long or how strong a vector is, making it crucial for differentiating between vectors that point in the same direction but vary in length.

Vector Direction

Understanding vector direction involves recognizing where and in what direction a vector points in its space. Despite many vectors potentially sharing the same direction, each can differ in magnitude. To work with direction, it's helpful to use a unit vector, which simplifies the vector to its directional component by ensuring its length is 1.
The direction is maintained while the size or 'strength' of the vector is standardized. Take vector \( \vec{v} = \langle 1, -2, 2 \rangle \) again; by scaling down its magnitude to 1, you get a sense of where it points without affecting its magnitude.
For this, use the formula for the unit vector:\[ \vec{u} = \frac{1}{||\vec{v}||} \cdot \vec{v} \]Where \( \vec{u} \) represents the unit vector in the same direction as \( \vec{v} \). The unit vector retains the same direction as the original vector but is limited in magnitude, so it clearly depicts the direction.

Scaling Vectors

Scaling vectors is the key process to change a vector's magnitude without altering its direction. By multiplying a vector by a scalar — a real number — we effectively stretch or shrink its length. The original vector's direction remains fixed.

• If the scalar is greater than 1, the vector elongates.
• If the scalar is between 0 and 1, the vector contracts.
• If negative, the vector also reverses direction.

To create a unit vector \( \vec{u} \) from \( \vec{v} = \langle 1, -2, 2 \rangle \), you scale \( \vec{v} \) by the reciprocal of its magnitude (1/3, since the magnitude is 3):\[ \vec{u} = \left\langle \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right\rangle \]This process shows how vectors can be adjusted to desired magnitudes while showing the original direction intentions, demonstrating the flexible nature of vectors in various mathematical and physical contexts.