Problem 79 - Precalculus

Question

If \(u\) and \(v\) are unit vectors, determine the maximum and minimum value of \((-2 \mathbf{u}) \cdot(3 \mathbf{v})\).

Step-by-Step Solution

Answer

Maximum is 6; Minimum is -6.

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a}\cdot\mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \), where \( \theta \) is the angle between the two vectors.

Step 2: Write Expression in Dot Product Form

Given the vectors \((-2 \mathbf{u})\) and \((3 \mathbf{v})\), express the dot product as:\[(-2 \mathbf{u}) \cdot (3 \mathbf{v}) = (-2)(3)(\mathbf{u} \cdot \mathbf{v}) = -6 (\mathbf{u} \cdot \mathbf{v})\].

Step 3: Evaluate Magnitude and Direction

Since \( \mathbf{u} \) and \( \mathbf{v} \) are unit vectors, their magnitudes are 1. Thus, \(|\mathbf{u}| |\mathbf{v}| = 1\). The dot product reduces to \( \mathbf{u} \cdot \mathbf{v} = \cos \theta \).

Step 4: Determine Possible Values for \( \cos \theta \)

The value of \( \cos \theta \) ranges from -1 to 1. Hence, for unit vectors \( \mathbf{u} \cdot \mathbf{v} = \cos \theta \), the dot product can vary from -1 to 1.

Step 5: Calculate Maximum and Minimum Values

Substituting in the previous expression, the maximum occurs when \( \cos \theta = 1 \), giving:\[-6(1) = -6\].The minimum occurs when \( \cos \theta = -1 \), giving:\[-6(-1) = 6\].

Key Concepts

Unit Vectors

In the realm of vector mathematics, unit vectors play a pivotal role. Essentially, a unit vector is a vector with a magnitude (or length) of exactly 1. This characteristic makes them particularly handy when dealing with directions because they point along an axis without emphasizing length. For example, common unit vectors include \(i\), \(j\), and \(k\) in three-dimensional space, corresponding to the x, y, and z axes, respectively.

Unit vectors are often used to specify directions in complex calculations, such as those involving the dot product. In our specific problem, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors, meaning their magnitudes are each 1. This simplifies computations since the product of their magnitudes equals 1 (\(|\mathbf{u}| |\mathbf{v}| = 1\)). This simplification allows us to focus entirely on the cosine of the angle between the vectors when calculating their dot product.

Cosine of Angle Between Vectors

When calculating the dot product, the cosine of the angle between two vectors is a crucial component. The dot product formula is given by \[\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\] where \(\theta\) is the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\).

For unit vectors, where each vector has a magnitude of 1, the formula simplifies to:

• \(\mathbf{a}\cdot\mathbf{b} = \cos \theta\)

Thus, the dot product reflects directly the cosine of the angle between the vectors. As the angle changes, the value of \(\cos \theta\) itself varies between -1 and 1:

• If the vectors point in the same direction (\(\theta = 0\)), \(\cos \theta = 1\).
• If the vectors are perpendicular (\(\theta = 90^\circ\)), \(\cos \theta = 0\).
• If the vectors point in opposite directions (\(\theta = 180^\circ\)), \(\cos \theta = -1\).

In the exercise at hand, understanding how \(\cos \theta\) behaves is essential for determining the maximum and minimum values of the dot product, since it dictates the result of the entire expression.

Maximum and Minimum Values of Dot Product

The dot product between two vectors can have various values depending on the angle between them. For unit vectors, since their magnitudes are 1, the value of their dot product equation narrows down to just the \(\cos \theta\) term.

The exercise asks for the maximum and minimum values of \((-2 \mathbf{u}) \cdot (3 \mathbf{v})\). Expressing this using the dot product, we find: \[(-2 \mathbf{u}) \cdot (3 \mathbf{v}) = -6 (\mathbf{u} \cdot \mathbf{v}) = -6 \cos \theta\] Given that \(\cos \theta\) ranges from -1 to 1, we must evaluate where this function reaches its bounds:

• At \(\cos \theta = 1\), the dot product becomes \(-6(1) = -6\).
• At \(\cos \theta = -1\), the expression becomes \(-6(-1) = 6\).

Thus, the minimum value of the dot product, \(-6\), occurs when the vectors are parallel in the same direction, and the maximum value, 6, occurs when they point in opposite directions. Knowing these extremities helps paint a full picture of how vectors interact through dot products as their orientations change.