Question

Find a vector orthogonal to the given vectors. $$\langle 6,-2,4\rangle \text { and }\langle 1,2,3\rangle$$

Short answer

Question: Calculate the vector orthogonal to the given vectors A = 〈6, -2, 4〉 and B = 〈1, 2, 3〉. Answer: The orthogonal vector is 〈-14, -14, 14〉.

Step-by-step solution

Write down the given vectors

We are given the following vectors: Vector A: \(\langle 6, -2, 4 \rangle\) Vector B: \(\langle 1, 2, 3 \rangle\)

Compute the cross product

To calculate the cross product of the given vectors, we can use the following equation: $$\vec{C} = \vec{A} \times \vec{B} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$$ where \(\vec{A} = \langle a_1, a_2, a_3 \rangle\), \(\vec{B} = \langle b_1, b_2, b_3 \rangle\), and \(\vec{C}\) is the resulting cross product.

Substitute and compute

Now substitute the given values into the equation: $$\vec{C} = \langle (-2)(3) - (4)(2), (4)(1) - (6)(3), (6)(2) - (-2)(1) \rangle$$ Calculate each component of the resulting vector: $$\vec{C} = \langle -6 - 8, 4 - 18, 12 + 2 \rangle$$ Get the resulting orthogonal vector: $$\vec{C} = \langle -14, -14, 14 \rangle$$ So, the vector orthogonal to both given vectors is \(\langle -14, -14, 14 \rangle\).

Key concepts

Cross Product

When you hear about the cross product, it's all about a special operation involving vectors. This operation is used in three-dimensional space to find a vector that is perpendicular to the plane formed by the original two vectors. The cross product of two vectors, say \( \vec{A} \) and \( \vec{B} \), is denoted as \( \vec{A} \times \vec{B} \). The result is a new vector that both points perpendicular and has a magnitude directly related to the area of the parallelogram spanned by \( \vec{A} \) and \( \vec{B} \).

• The cross product is anti-commutative, meaning \( \vec{A} \times \vec{B} = - (\vec{B} \times \vec{A}) \).
• It's computed using the determinant of a matrix formed by unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of the vectors.

This operation is significant as it helps in various applications such as physics, engineering, and computer graphics for finding normals to surfaces.

Orthogonal Vectors

Two vectors are said to be orthogonal when they are at a 90-degree angle to each other. This is another way of saying they are perpendicular. When vectors are orthogonal, their dot product equals zero; however, sometimes we identify orthogonal vectors via the cross product, especially in three-dimensional space. The cross product itself is a vector orthogonal to both original vectors.

• If \( \vec{C} \) is orthogonal to both \( \vec{A} \) and \( \vec{B} \), then \( \vec{C} \cdot \vec{A} = 0 \) and \( \vec{C} \cdot \vec{B} = 0 \).
• In geometry, orthogonal vectors form the axes of coordinate systems, like the usual X, Y, and Z axes in three-dimensional space.

Understanding orthogonal vectors is important in fields like linear algebra and data science because they help in simplifying problems or decomposing spaces into understandable parts.

Vector Operations

Vector operations include a variety of processes like addition, subtraction, scalar multiplication, dot product, and cross product. Each of these plays distinct roles in manipulating vectors, which are essential entities in physics, engineering, and computer science.

• Addition and Subtraction: For vectors \( \vec{A} \) and \( \vec{B} \), you add or subtract corresponding components.
• Scalar Multiplication: Involves multiplying every component of a vector by a scalar, changing the magnitude but not the direction.
• Dot Product: Gives a scalar, calculated by multiplying corresponding components and summing them. It's useful to find if vectors are orthogonal by resulting in zero.
• Cross Product: Yields a vector as discussed, with applications in finding orthogonal vectors.

Proficiency with vector operations underpins much of modern mathematics and physics, providing tools necessary for tasks ranging from game development to robotics and beyond.