Question

Parallel and perpendicular vectors 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

Question: What are the values of \(a\) such that the vector \(\mathbf{u} = \langle a, 5 \rangle\) is: a. Parallel to \(\mathbf{v} = \langle 2, 6 \rangle\) b. Perpendicular to \(\mathbf{v} = \langle 2, 6 \rangle\) Answer: a. To make \(\mathbf{u}\) parallel to \(\mathbf{v}\), \(a = \frac{5}{3}\). b. To make \(\mathbf{u}\) perpendicular to \(\mathbf{v}\), \(a = -15\).

Step-by-step solution

Set up the equation for parallel vectors

If two vectors are parallel, they are scalar multiples of each other. So we can write the equation as: $$\mathbf{u} = k\mathbf{v}$$ where \(k\) is a scalar constant.

Substitute the given components and find the value of \(a\)

Substitute the given components of \(\mathbf{u}\) and \(\mathbf{v}\): $$\langle a, 5 \rangle = k\langle 2, 6\rangle$$ $$\langle a, 5 \rangle = \langle 2k, 6k \rangle$$ Now, equate the corresponding components and solve for \(a\): $$a = 2k$$ $$5 = 6k$$ Dividing the second equation by 6, we get: $$k =\frac{5}{6}$$ Now, substitute the value of \(k\) in the first equation: $$a = 2 \left(\frac{5}{6}\right)$$ $$a = \frac{5}{3}$$ So the value of \(a\) such that \(\mathbf{u}\) is parallel to \(\mathbf{v}\) is \(a = \frac{5}{3}\). #Part b: Perpendicular Vectors#

Set up the equation for perpendicular vectors

If two vectors are perpendicular, their dot product is equal to 0. So we can write the equation as: $$\mathbf{u} \cdot \mathbf{v} = 0$$

Substitute the given components and find the value of \(a\)

Substitute the given components of \(\mathbf{u}\) and \(\mathbf{v}\): $$\langle a, 5 \rangle \cdot \langle 2, 6 \rangle = 0$$ Now, calculate the dot product: $$(a)(2) + (5)(6) = 0$$ Solve for \(a\): $$2a + 30 = 0$$ $$2a = -30$$ $$a = -15$$ So the value of \(a\) such that \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\) is \(a = -15\).

Key concepts

Parallel Vectors

Parallel vectors are those that have the same or exact opposite direction. These vectors can be thought of as being just scaled versions of each other. In mathematics, if two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel, there exists a scalar \(k\) such that \(\mathbf{u} = k \mathbf{v}\). This means if one vector is multiplied by a scalar, it will result in the other vector.

To determine if two vectors are parallel, you can check if their components satisfy this relationship. For example, given vectors \(\mathbf{u} = \langle a, 5 \rangle\) and \(\mathbf{v} = \langle 2, 6 \rangle\), you set\( \langle a, 5 \rangle = k \langle 2, 6 \rangle \).

• The first component gives: \(a = 2k\)
• The second component gives: \(5 = 6k\)

From here, solve for \(k\) and substitute back to find \(a\). Parallel vectors are commonly seen in physics to describe motion in straight lines.

Perpendicular Vectors

Perpendicular vectors intersect at a right angle and have an interesting mathematical property: the dot product of two perpendicular vectors is zero. The dot product is calculated as the sum of the products of the vectors' corresponding components.

For vectors \(\mathbf{u} = \langle a, 5 \rangle\) and \(\mathbf{v} = \langle 2, 6 \rangle\), we find if they are perpendicular by setting \(\mathbf{u} \cdot \mathbf{v} = 0\). This results in the equation:

• \( (a)(2) + (5)(6) = 0 \)
• Simplifying, we have: \( 2a + 30 = 0 \)
• Solving gives: \( a = -15 \)

When vectors are perpendicular, they are used in many applications, such as to determine orthogonal components in physics or engineering.

Dot Product

The dot product is a way of multiplying two vectors to result in a scalar value. It's a bridge between algebra and geometry, providing useful insights into the orientations of two vectors.

The formula for the dot product of vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\) is:

• \(\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2\)

For our vectors, \(\mathbf{u} = \langle a, 5 \rangle\) and \(\mathbf{v} = \langle 2, 6 \rangle\), this becomes \( a \cdot 2 + 5 \cdot 6 = 0 \) to check for perpendicularity.

The dot product helps determine angles between vectors and the projection of one vector onto another. When two vectors have a dot product of zero, they are perpendicular. Understanding this product is essential for tasks ranging from calculating work done by a force in physics to manipulating data in computer graphics.