Selftest: Dot product

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Exercises for chapter Vector algebra | Article: Dot product

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1. What is the result of dor product of $\vec{\mathbf{a}}$ and $\vec{\mathbf{b}}$?

	$\begin{pmatrix} 3 \\ 5 \end{pmatrix}$
	1,5
	15
→	The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha$. The angle is 0° and so the cosine is 1. Further information: see Dot product.

2. What is the result of dor product of $\vec{\mathbf{a}}$ and $\vec{\mathbf{b}}$?

	$\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
	15
	0
→	The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha$. The angle is 90° and so the cosine is 0. Further information: see Dot product.

3. What is the result of dor product of $\vec{\mathbf{a}}$ and $\vec{\mathbf{b}}$?

	$\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
	-15
	15
→	The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha$. The angle is 180° and so the cosine is -1. Further information: see Dot product.

4. Please solve the following exercise:

$\begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}\cdot \begin{pmatrix} -3 \\ 6 \\ 4 \end{pmatrix}=$

→ There are two possibilities to compute the dot product. Either the component representation $\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3$ is used or the angle between the vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha$.

5. Please solve the following exercise:

$\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\cdot \begin{pmatrix} -4 \\ 1 \\ -6 \end{pmatrix}=$

→ There are two possibilities to compute the dot product. Either the component representation $\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3$ is used or the angle between the vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha$.

6. Please solve the following exercise:

$\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=$

→ There are two possibilities to compute the dot product. Either the component representation $\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3$ is used or the angle between the vectors: $\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha$.

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