Notes
Orthogonal Vectors
Dot Product \(\vec{u} \cdot \vec{v}=0\)
Practice Problems
\(\textbf{1)}\) Are \( \vec{u}=(1,2,3) \) and \( \vec{v}=(8,-4,0) \) orthogonal?
\(\textbf{2)}\) Are \( \vec{u}=(1,5,8) \) and \( \vec{v}=(3,-5,1) \) orthogonal?
\(\textbf{3)}\) Are \( \vec{u}=-5,-4,10) \) and \( \vec{v}=4,5,4) \) orthogonal?
\(\textbf{4)}\) Are \( \vec{u}=(-2,3,5) \) and \( \vec{v}=(5,4,2) \) orthogonal?
Challenge Problem
\(\textbf{5)}\) Find k so that \( \vec{u}=(2,3,4) \) and \( \vec{v}=(-5,k,1) \) are orthogonal.
See Related Pages\(\)
\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)
\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)
\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)
\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)
\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)
\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)
In Summary
Orthogonal vectors are two or more vectors that are perpendicular to each other. In mathematical terms, orthogonal vectors are defined as vectors that have a dot product of zero. This means that if you multiply the components of the two vectors together and add them up, the result is zero.
Orthogonal vectors are typically studied in linear algebra classes, and are very important in math, engineering and physics.
A fun fact about orthogonal vectors is that they are often used to describe the orientation of planes in three-dimensional space. For example, the three axes of a coordinate system (x, y, and z) are all orthogonal to each other.
Some related topics to orthogonal vectors include linear independence, basis vectors, and projection.