Cross Product

Notes

Cross Product Notes

Cross Product Notes

Determinate of 3x3 Matrix Notes

Practice Problem

\(\textbf{1)}\) Find \(\,\,\vec{r}\times\vec{s}\,\,\) where \(\,\,\vec{r}=2\vec{i}+5\vec{j}-1\vec{k}\,\,\) and \(\,\,\vec{s}=3\vec{i}-4\vec{j}+6\vec{k}\)Link to Youtube Video Solving Question Number 1

 

\(\textbf{2)}\) Find \(\,\,\vec{u}\times\vec{v}\,\,\) where \(\,\,\vec{u}=\langle2,5,3\rangle\,\,\) and \(\,\,\vec{v}=\langle6,-2,1\rangle\)

 

\(\textbf{3)}\) Find \(\,\,\vec{a}\times\vec{b}\,\,\) where \(\,\,\vec{a}=\langle1,3,4\rangle\,\,\) and \(\,\,\vec{b}=\langle-5,1,1\rangle\)
\(\,\,\,\,\,\, \left| {\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 3 & 4 \\ -5 & 1 & 1 \\\end{array} } \right|\) \(\,\,\,\,\,\,\vec{i} \left| {\begin{array}{cc} 3 & 4 \\ 1 & 1 \\\end{array} } \right| -\vec{j} \left| {\begin{array}{cc} 1 & 4 \\ -5 & 1 \\\end{array} } \right| +\vec{k} \left| {\begin{array}{cc} 1 & 3 \\ -5 & 1 \\\end{array} } \right| \) \(\,\,\,\,\,\,\vec{i} \left( (3)(1)-(4)(1) \right) -\vec{j} \left( (1)(1)-(4)(-5) \right) +\vec{k} \left( (1)(1)-(3)(-5) \right) \) \(\,\,\,\,\,\,\vec{i} \left( (3)-(4) \right) -\vec{j} \left( (1)-(-20) \right) +\vec{k} \left( (1)-(-15) \right) \) \(\,\,\,\,\,\,\vec{i} \left( 3-4 \right) -\vec{j} \left( 1+20 \right) +\vec{k} \left( 1+15 \right) \)

The answer is \(\vec{a}\times\vec{b}=\langle-1,-21,16\rangle\)

 

\(\textbf{4)}\) Find \(\,\,\vec{m}\times\vec{n}\,\,\) where \(\,\,\vec{m}=\langle1,2,3\rangle\,\,\) and \(\,\,\vec{n}=\langle1,0,0\rangle\)

 

\(\textbf{5)}\) Find \(\,\,\vec{u}\times\vec{v}\,\,\) where \(\,\,\vec{u}=\langle1,2,3\rangle\,\,\) and \(\,\,\vec{v}=\langle3,2,1\rangle\)

 

True or False

\(\textbf{6)}\) \(\,\,\vec{u}\times\vec{v}\,\,=\,\,\vec{v}\times\vec{u}\)

 

\(\textbf{7)}\) \(\,\,||\vec{u}\times\vec{v}||\,\,=\,\,||\vec{v}\times\vec{u}||\)

 

\(\textbf{8)}\) The cross product of two parallel vectors is zero.

 

\(\textbf{9)}\) The cross product of two vectors in two-dimensional space always results in a vector perpendicular to the plane containing the original vectors.

 

 

See Related Pages\(\)

\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)
\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)
\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)
\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)
\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)
\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Cross Product\(…\)
\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)

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