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133 - FML 3 - 18.1W e.g. 22

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Wize University Linear Algebra Textbook > Products of Vectors

Dot Product Properties

5 Activities

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How do you express the idea that two vectors v⃗\bcb{\vec{v}} and w⃗\bcb{\vec{w}} are perpendicular, mathematically?



More Dot Product Properties Questions:
133 - FML 3 - 18.1W e.g. 46
If ∣u⃗∣=2\bcb{|{\vec{u}|} = 2} and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}, where u⃗⋅v⃗=2\bcb{\vec{u} \cdot \vec{v} = 2}, find ∣u⃗−v⃗∣\bcb{|{\vec{u} - \vec{v} }|}.
133 - FML 3 - 18.1W e.g. 40
If u⃗=<1,−2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>} and v⃗=<2, −1, 3>\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}, find (2u⃗)⋅(3v⃗)\bcb{(2\vec{u})\cdot (3\vec{v})}.
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that u⃗, v⃗, w⃗\vec{u},\ \vec{v},\ \vec{w} are vectors in R2R^2, p⃗, q⃗, r⃗\vec{p},\ \vec{q},\ \vec{r} are vectors in R3R^3 and a, ba,\ b are scalars.
Which of the following are valid operations? (Select all that apply)
Practice: Dot and Cross Product Properties
Practice: Dot and Cross Product Properties
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ , \vec𝑣 , 𝑤⃗ are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following is/are true?
Dot Product Properties
Practice: Dot Product Properties
Let u⃗, v⃗,w⃗∈R3\vec{u},\ \vec{v}, \vec{w} \in \reals^3 be vectors and let c, d∈Rc,\ d \in \reals be scalars. Select all expressions that are valid.
133 - FML 3 - 18.1W e.g. 54
Find the value of k\bcb{k} such that the vector u⃗=<1,k,3>\bcb{\vec{u} = \left< 1, k, 3 \right>} is perpendicular to the vector v⃗=<2,−1,k>\bcb{\vec{v} = \left< 2, -1, k\right>}.
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that u⃗, v⃗, w⃗\vec{u},\ \vec{v},\ \vec{w} are vectors in R2R^2, p⃗, q⃗, r⃗\vec{p},\ \vec{q},\ \vec{r} are vectors in R3R^3 and a, ba,\ b are scalars.
Which of the following are valid operations? (Select all that apply)
Find the value of 𝑘 such that the vectors 𝑢⃗=(1,𝑘,2𝑘)𝑢⃗=(1,𝑘,2𝑘) 𝑎𝑛𝑑 𝑣⃗=(3,1,0)𝑣⃗=(3,1,0) are orthogonal.
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If u⃗=(−2,3,−1),  v⃗=(−1,−2,3)\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right), find
a) u⃗ ∙ v⃗\vec{u}\ \bullet\ \vec{v}
Given that u→=(5,−1,2,k)\overrightarrow{u}=\left(5,-1,2,k\right)and v→=(−10,2,−4,6)\overrightarrow{v}=\left(-10,2,-4,6\right), which of the following statements is/are true?
If k=3k=3, then u→\overrightarrow{u}and v→\overrightarrow{v}are parallel/collinear
If k=−10k=-10, then u→\overrightarrow{u}and v→\overrightarrow{v}are orthogonal/perpendicular
Given that u→\overrightarrow{u}, v→\overrightarrow{v}and w→\overrightarrow{w}are vectors in R3\mathbb{R}^3and kkis a scalar (constant). Which of the following operations are defined (possible)?
u→⋅(v→×w→)\overrightarrow{u}\cdot(\overrightarrow{v}\times\overrightarrow{w})
u→+(v→⋅w→)\overrightarrow{u}+(\overrightarrow{v}\cdot\overrightarrow{w})
Dot Product Properties
Find all values of kk such that the vectors u=(−1,2,k,0)\bm{u}=(-1,2,k,0) and v=(k,−6,6,0)\bm{v}=(k,-6,6,0) are orthogonal.