Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Notes

Practice Problem

\(\textbf{1)}\) Find \(\,\,\vec{r}\times\vec{s}\,\,\) where \(\,\,\vec{r}=2\vec{i}+5\vec{j}-1\vec{k}\,\,\) and \(\,\,\vec{s}=3\vec{i}-4\vec{j}+6\vec{k}\)

Show Answer

The answer is \(\vec{r}\times\vec{s}=26\vec{i}-15\vec{j}-23\vec{k}\)

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 2 & 5 & -1 \\ 3 & -4 & 6 \\\end{array} } \right|\)


\(\,\,\,\,\,\,\vec{i} \left| {\begin{array}{cc} 5 & -1 \\ -4 & 6 \\\end{array} } \right| -\vec{j} \left| {\begin{array}{cc} 2 & -1 \\ 3 & 6 \\\end{array} } \right| +\vec{k} \left| {\begin{array}{cc} 2 & 5 \\ 3 & -4 \\\end{array} } \right| \)


\(\,\,\,\,\,\,\vec{i} \left( (5)(6)-(-1)(-4) \right) -\vec{j} \left( (2)(6)-(-1)(3) \right) +\vec{k} \left( (2)(-4)-(3)(5) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( (30)-(4) \right) -\vec{j} \left( (12)-(-3) \right) +\vec{k} \left( (-8)-(15) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( 26 \right) -\vec{j} \left( 15 \right) +\vec{k} \left( -23 \right) \)


\(\,\,\,\,\,\,\)The answer is \(\vec{r}\times\vec{s}=26\vec{i}-15\vec{j}-23\vec{k}\)

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

Show Answer

The answer is \(\vec{u}\times\vec{v}=\langle11,16,-34\rangle\)

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 2 & 5 & 3 \\ 6 & -2 & 1 \\\end{array} } \right|\)


\(\,\,\,\,\,\,\vec{i} \left| {\begin{array}{cc} 5 & 3 \\ -2 & 1 \\\end{array} } \right| -\vec{j} \left| {\begin{array}{cc} 2 & 3 \\ 6 & 1 \\\end{array} } \right| +\vec{k} \left| {\begin{array}{cc} 2 & 5 \\ 6 & -2 \\\end{array} } \right| \)


\(\,\,\,\,\,\,\vec{i} \left( (5)(1)-(3)(-2) \right) -\vec{j} \left( (2)(1)-(3)(6) \right) +\vec{k} \left( (2)(-2)-(5)(6) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( (5)-(-6) \right) -\vec{j} \left( (2)-(18) \right) +\vec{k} \left( (-4)-(30) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( 11 \right) -\vec{j} \left( -16 \right) +\vec{k} \left( -34 \right) \)


\(\,\,\,\,\,\,\)The answer is \(\vec{u}\times\vec{v}=\langle11,16,-34\rangle\)

\(\textbf{3)}\) Find \(\,\,\vec{a}\times\vec{b}\,\,\) where \(\,\,\vec{a}=\langle1,3,4\rangle\,\,\) and \(\,\,\vec{b}=\langle-5,1,1\rangle\)

Show Answer

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

Show Work

\(\,\,\,\,\,\, \left| {\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 3 & 4 \\ -5 & 1 & 1 \\\end{array} } \right|\)


\(\,\,\,\,\,\,\vec{i} \left| {\begin{array}{cc} 3 & 4 \\ 1 & 1 \\\end{array} } \right| -\vec{j} \left| {\begin{array}{cc} 1 & 4 \\ -5 & 1 \\\end{array} } \right| +\vec{k} \left| {\begin{array}{cc} 1 & 3 \\ -5 & 1 \\\end{array} } \right| \)


\(\,\,\,\,\,\,\vec{i} \left( (3)(1)-(4)(1) \right) -\vec{j} \left( (1)(1)-(4)(-5) \right) +\vec{k} \left( (1)(1)-(3)(-5) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( (3)-(4) \right) -\vec{j} \left( (1)-(-20) \right) +\vec{k} \left( (1)-(-15) \right) \)


\(\,\,\,\,\,\,\vec{i} \left( 3-4 \right) -\vec{j} \left( 1+20 \right) +\vec{k} \left( 1+15 \right) \)


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 \(\,\,\vec{n}=\langle1,0,0\rangle\)

Show Answer

The answer is \(\vec{m}\times\vec{n}=\langle0,3,-2\rangle\)

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

Show Answer

The answer is \(\vec{u}\times\vec{v}=\langle-4,8,-4\rangle\)

True or False

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

True or False

The answer is False

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

True or False

The answer is True

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

True or False

The answer is True

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

True or False

The answer is True

See Related Pages\(\)

\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)

\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)

\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)

\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)

\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)

\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!