The Dot Product
1. Let \({\bf a} = \langle 1, -2, 3
\rangle \), \({\bf b } = \langle 4, 0, -1 \rangle \), and
\({\bf c} = \langle 2, 4, -5 \rangle \). Find (i) \(
({\bf a \cdot b})\bf c\), (ii) \({\bf b}\cdot (2\bf a) \), (iii)
\(\|\bf c\|^2 \).
SOLUTION
2. Let
\(\overrightarrow{v}=3\overrightarrow{i}-5\overrightarrow{j}\) and \(
\overrightarrow{w}= -2\overrightarrow{i} + 3\overrightarrow{j}\). Find
(a) \( ||\overrightarrow{v}-\overrightarrow{w}||\), and (b)
\(||\overrightarrow{v}||+||\overrightarrow{w}||\). SOLUTION
3. Let \(\overrightarrow{v}=2\overrightarrow{i} -
3\overrightarrow{j}\) and \(\overrightarrow{w}= -3\overrightarrow{i}+
4\overrightarrow{j}\). Find (a) \(
||\overrightarrow{v}-\overrightarrow{w}||\), and (b)
\(||\overrightarrow{v}||-||\overrightarrow{w}||\).
SOLUTION
4. Find the angle between the given vectors, and state whether they
are parallel, orthogonal or neither.
SOLUTION
\(\overrightarrow{u}=2\overrightarrow{i}+
\overrightarrow{j}, \quad \overrightarrow{v}=\overrightarrow{i} -
2\overrightarrow{j}\)
5. Find the angle between the given vectors, and state whether they
are parallel, orthogonal or neither.
SOLUTION
\(\overrightarrow{u}=2\overrightarrow{i}-5
\overrightarrow{j}, \quad \overrightarrow{v}=4\overrightarrow{i} -
10\overrightarrow{j}\)
6. Let \({\bf a} = \langle 3, -1, 2 \rangle \) and \({\bf b} =
\langle 1, -1, -2 \rangle \) be given. Find the measure of the angle
between them. SOLUTION
7. Let \({\bf a} = \langle 1, 2, 2\rangle \). Find the measure of
the angle (in radians) formed by \(\bf a \) and \(\bf j \).
SOLUTION
8. Let \({\bf u} = \langle 5, -3, 2 \rangle \) and \({\bf v} =
\langle 2, 2, -1 \rangle \) be given. Find the measure of the angle
between them.
SOLUTION
9. Find the scalar projection, and vector projection of
\({\bf b} = \langle 1, 3, -5 \rangle \) onto \({\bf a} =
\langle 2, -2, 3 \rangle \).
SOLUTION