Question

A vector \(\vec{v}\) is given. Give two vectors that are orthogonal to \(\vec{v}\). \(\vec{v}=\langle 1,1,1\rangle\)

Short answer

Two orthogonal vectors are \(\langle 1, 1, -2 \rangle\) and \(\langle 1, -1, 0 \rangle\).

Step-by-step solution

Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. For vectors \(\vec{v} = \langle 1, 1, 1 \rangle\) and \(\vec{u} = \langle a, b, c \rangle\), the dot product is given by \(1 \cdot a + 1 \cdot b + 1 \cdot c\). To ensure orthogonality, this sum must equal zero: \(a + b + c = 0\).

Choose Values for Vector \(\vec{u_1}\)

We need values for \(a\), \(b\), and \(c\) such that the equation \(a + b + c = 0\) holds. Choosing \(a = 1\), \(b = 1\), and \(c = -2\) satisfies this condition, so one possible vector is \(\vec{u_1} = \langle 1, 1, -2 \rangle\).

Verify Orthogonality of \(\vec{u_1}\)

Calculate the dot product of \(\vec{v} = \langle 1, 1, 1 \rangle\) and \(\vec{u_1} = \langle 1, 1, -2 \rangle\): \(1 \cdot 1 + 1 \cdot 1 + 1 \cdot (-2) = 1 + 1 - 2 = 0\). Since the dot product is zero, \(\vec{u_1}\) is orthogonal to \(\vec{v}\).

Choose Values for Vector \(\vec{u_2}\)

Similarly, we can choose another set of values for \(a\), \(b\), and \(c\). Let's take \(a = 1\), \(b = -1\), and \(c = 0\) for a second vector \(\vec{u_2}\). This gives us \(\vec{u_2} = \langle 1, -1, 0 \rangle\).

Verify Orthogonality of \(\vec{u_2}\)

Check the dot product of \(\vec{v} = \langle 1, 1, 1 \rangle\) and \(\vec{u_2} = \langle 1, -1, 0 \rangle\): \(1 \cdot 1 + 1 \cdot (-1) + 1 \cdot 0 = 1 - 1 + 0 = 0\). Since the dot product is zero, \(\vec{u_2}\) is also orthogonal to \(\vec{v}\).

Key concepts

Dot Product

The dot product is a way to multiply two vectors and get a scalar, which is just a fancy word for a single number. It's a key operation in vector mathematics often used to see how two vectors relate to one another.
To calculate the dot product between two vectors, say \( \vec{v} = \langle x_1, y_1, z_1 \rangle \) and \( \vec{u} = \langle a, b, c \rangle \), we multiply their corresponding components and then sum them up:

• Multiply \( x_1 \) by \( a \)
• Multiply \( y_1 \) by \( b \)
• Multiply \( z_1 \) by \( c \)

Then add these products together to get the dot product. For example, if \( \vec{v} = \langle 1, 1, 1 \rangle \) and \( \vec{u} = \langle a, b, c \rangle \), the dot product is calculated as \( 1 \cdot a + 1 \cdot b + 1 \cdot c = a + b + c \).
If these vectors are orthogonal, this sum will be zero, meaning they meet at a right angle.

Vector Mathematics

Vectors are like magical arrows that can tell you not only how far but also in which direction to go. They are fundamental in describing any point in space, physics, graphics, and much more. Every vector has a magnitude and direction.

• A vector in the plane is typically represented as \( \langle x, y \rangle \), while in space, it's \( \langle x, y, z \rangle \).
• The length or magnitude of a vector \( \vec{a} = \langle x, y, z \rangle \) is measured as \( \sqrt{x^2 + y^2 + z^2} \).
• You can add and subtract vectors by dealing with their individual components: for instance adding \( \langle x_1, y_1 \rangle \) and \( \langle x_2, y_2 \rangle \) gives \( \langle x_1 + x_2, y_1 + y_2 \rangle \).

These operations allow vectors to represent complex ideas, such as force, velocity, and more. Understanding these basics is crucial for delving deeper into vector mathematics.

Orthogonality Condition

Orthogonal vectors are perpendicular, which is a neat way of saying they meet at a right angle. The magic rule here is that their dot product equals zero. This is called the orthogonality condition.
To find if two vectors \( \vec{v} = \langle x_1, y_1, z_1 \rangle \) and \( \vec{u} = \langle a, b, c \rangle \) are orthogonal, compute their dot product. If \( x_1 \cdot a + y_1 \cdot b + z_1 \cdot c = 0 \), they’re orthogonal.
This concept is super helpful, especially in geometry and physics, as it plays a role in determining angles and orientations. Also, in higher math, it's vital in forming orthogonal bases or simplifying equations involving transformations.
In our example with \( \vec{v} = \langle 1, 1, 1 \rangle \), vectors like \( \langle 1, 1, -2 \rangle \) and \( \langle 1, -1, 0 \rangle \) fulfill \( 1 \cdot a + 1 \cdot b + 1 \cdot c = 0 \) because their calculations give a zero dot product, thus guaranteeing orthogonality.