9709 P3 · Notes & worked examples
Check a final answer, then explain the reasoning on paper. Guest progress stays in this browser; an account saves future checks without overwriting your first result.
Recall
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}2\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\3\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find u·v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}3\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\1\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find u·v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Method
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}3\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\1\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find \(\displaystyle |u|^{2}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}4\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\2\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find \(\displaystyle |u|^{2}\).
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Apply
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}4\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\2\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find \(\displaystyle \cos ^{2}\theta\) for the angle between u and v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}1\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\3\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find \(\displaystyle \cos ^{2}\theta\) for the angle between u and v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Check understanding
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}1\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\3\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find k so that u+(k,0,0) is perpendicular to v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.
Fresh review variant
Maths4U Original · Short practice
\(\mathbf u=\left[\begin{matrix}2\\2\\3\end{matrix}\right],\quad\mathbf v=\left[\begin{matrix}2\\1\\1\end{matrix}\right]\). Let θ be the angle between these two vectors in [0,π].
Find k so that u+(k,0,0) is perpendicular to v.
Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.