Question

Let \(\mathbf{u}=\langle a, 5\rangle\) and \(\mathbf{v}=\langle 2,6\rangle\) a. Find the value of \(a\) such that \(\mathbf{u}\) is parallel to \(\mathbf{v}\) b. Find the value of \(a\) such that \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\)

Short answer

Answer: For the given vectors, \(a=\frac{5}{3}\) when they are parallel, and \(a=-15\) when they are perpendicular.

Step-by-step solution

Write the given vectors in terms of components

Given, \(\mathbf{u}=\langle a, 5\rangle\) and \(\mathbf{v}=\langle 2,6\rangle\).

Condition for parallel vectors

For vectors \(\mathbf{u}\) and \(\mathbf{v}\) to be parallel, they need to be scalar multiples of each other. This means we can write \(\mathbf{u} = k\mathbf{v}\), where \(k\) is a constant. For the components, we have the following relationship: $$ a = k \cdot 2 \quad\text{and}\quad 5 = k \cdot 6 $$

Solve for \(a\) for parallel vectors

We have the equations above: $$ a = 2k \\ 5 = 6k $$ From the second equation, we can find the value of \(k\): $$ k = \frac{5}{6} $$ Now, substitute the value of \(k\) in the first equation: $$ a=2(\frac{5}{6})=\frac{10}{6}=\frac{5}{3} $$ So, \(a=\frac{5}{3}\) for \(\mathbf{u}\) and \(\mathbf{v}\) to be parallel.

Condition for perpendicular vectors

For vectors \(\mathbf{u}\) and \(\mathbf{v}\) to be perpendicular, their dot product should be zero. The dot product is given by: $$ \mathbf{u} \cdot \mathbf{v} = a \cdot 2 + 5 \cdot 6 = 0 $$

Solve for \(a\) for perpendicular vectors

We have the equation from Step 4: $$ 2a + 5 \cdot 6 = 0 $$ Now, solve for \(a\): $$ 2a +30=0 \\ 2a = -30 \\ a = -15 $$ So, \(a=-15\) for \(\mathbf{u}\) and \(\mathbf{v}\) to be perpendicular.

Summary

The value of \(a\) for \(\mathbf{u}\) to be parallel to \(\mathbf{v}\) is \(\frac{5}{3}\), and the value of \(a\) for \(\mathbf{u}\) to be perpendicular to \(\mathbf{v}\) is \(-15\).

Key concepts

Parallel Vectors

Vectors are fundamental in calculus and geometry, serving as a representation of direction and magnitude in space. Parallel vectors are vectors that have the same or exactly opposite direction. An essential understanding is that parallel vectors must be scalar multiples of each other. This means if we have two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), and we say that \(\mathbf{u}\) is parallel to \(\mathbf{v}\), then there exists a scalar value \(k\) such that \(\mathbf{u} = k\mathbf{v}\).

The components of these vectors maintain a consistent proportional relationship. For the exercise at hand, the value of \(a\) was determined by setting the components of vector \(\mathbf{u}\) as multiples of \(\mathbf{v}\)'s corresponding components. By finding the scalar \(k\) and applying it to one component, we discovered the necessary value for \(a\) to ensure the vectors are parallel.

Perpendicular Vectors

Perpendicular vectors, in contrast to parallel vectors, are vectors that meet at a right angle (90 degrees). In a two-dimensional plane, this means that one vector is vertical while the other is horizontal if they are perpendicular. A critical concept here is the dot product, also known as the scalar product. When two vectors are perpendicular, their dot product equals zero, \(\mathbf{u} \cdot \mathbf{v} = 0\). The dot product for two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) with components \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\) is calculated as \(a \cdot c + b \cdot d\).

In the given exercise, using the dot product property allowed the determination of the value for \(a\), where vector \(\mathbf{u}\) is perpendicular to vector \(\mathbf{v}\), signifying the importance of this calculation in finding mutually orthogonal vector pairs.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation combines the product of the vectors' magnitudes with the cosine of the angle between them. In its simplest form, for two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) with components \(\langle a, b \rangle\) and \(\langle c, d \rangle\), the dot product is given by the formula \(\mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d\).

Understanding the dot product is crucial for determining when vectors are perpendicular, as seen in the exercise. A dot product of zero is a clear indicator that vectors are orthogonal. Additionally, the magnitude of the dot product can show how much one vector extends in the direction of another, making it a powerful tool in vector analysis.

Scalar Multiple

In vector operations, a scalar multiple refers to a vector whose elements have been multiplied by a scalar value. Simply put, if we have a vector \(\mathbf{v}\), and a scalar \(k\), the scalar multiple of \(\mathbf{v}\) by \(k\) is a new vector \(k\mathbf{v}\), where each component of \(\mathbf{v}\) has been multiplied by \(k\). Scalar multiplication does not change the direction of a vector (unless \(k\) is negative, in which case the direction is reversed), but it does change the magnitude.

In the context of the exercise, the concept of scalar multiple was essential in finding the value of \(a\) for which vector \(\mathbf{u}\) is parallel to \(\mathbf{v}\). By examining the relationship between the corresponding components of the vectors, the correct scalar \(k\) was found, providing an effective method for analyzing and understanding vectors in multidimensional space.