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Practice Question: Vector Properties

Related Topics

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product Properties

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Practice Question: Vector Properties

If u⃗, v⃗, w⃗  ∈ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)

i.) u⃗⋅v⃗‖w⃗‖\frac{\vec{u}\cdot\vec{v}}{‖\vec{w}‖}
ii.) 𝑢⃗+(𝑣⃗×𝑤⃗)𝑢⃗+(𝑣⃗\times𝑤⃗)
iii.) 𝑢⃗−‖𝑣⃗×𝑤⃗‖𝑢⃗−‖𝑣⃗\times𝑤⃗‖
iv.) (u⃗⋅v⃗)×w⃗\left(\vec{u}\cdot\vec{v}\right)\times\vec{w}
v.)(𝑢⃗×𝑣⃗)⋅𝑤⃗(𝑢⃗\times𝑣⃗)\cdot𝑤⃗

i.) Since the vectors have the same number of components, u⃗⋅v⃗\vec{u}\cdot\vec{v} is defined. Also, ∥w⃗∥\Vert \vec w \Vert is a scalar, so u⃗⋅v⃗‖w⃗‖\frac{\vec{u}\cdot\vec{v}}{‖\vec{w}‖} is defined.

ii.) v⃗×w⃗\vec v \times \vec w is a vector in R3R^3. Therefore, 𝑢⃗+(𝑣⃗×𝑤⃗)𝑢⃗+(𝑣⃗\times𝑤⃗) is defined.

iii.) ‖𝑣⃗×𝑤⃗‖‖𝑣⃗\times𝑤⃗‖ is a scalar. Since we cannot subtract a vector and a scalar, 𝑢⃗−‖𝑣⃗×𝑤⃗‖𝑢⃗−‖𝑣⃗\times𝑤⃗‖ is not defined.

iv.) (u⃗⋅v⃗)\left(\vec{u}\cdot\vec{v}\right) is a scalar. Since we cannot take the cross product between a scalar and a vector, (u⃗⋅v⃗)×w⃗\left(\vec{u}\cdot\vec{v}\right)\times\vec{w} is not defined.

v.) (𝑢⃗×𝑣⃗)(𝑢⃗\times𝑣⃗) produces a vector in R3R^3. We can find the dot product between two vectors that have the same number of components. Therefore, (𝑢⃗×𝑣⃗)⋅𝑤⃗(𝑢⃗\times𝑣⃗)\cdot𝑤⃗ is defined.
More Cross Product Properties Questions:
Practice Question: Dot and Cross Product Properties
Practice Question: Dot and Cross Product Properties
Given that u⃗=(1,−2,0,1)\vec{u}=\left(1,-2,0,1\right) and 𝑣⃗\vec𝑣 =(0,1,2,1),=(0,1,2,1), which of the following is true?
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ ,\vec{𝑣} , \vec𝑤 are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i) (𝑤⃗×𝑐𝑢⃗)+𝑑𝑣⃗(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
ii) (w⃗×𝑐𝑢⃗)⋅𝑑𝑣(\vec{w}×𝑐𝑢⃗ ) \cdot 𝑑𝑣
Practice Question: Properties of Dot and Cross Products
Practice Question: Properties of Dot and Cross Products
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ ,\vec{𝑣} , \vec𝑤 are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i) (𝑤⃗×𝑐𝑢⃗)+𝑑𝑣⃗(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
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?
Practice: Cross Product (1)
Do the vectors u⃗= <3, −1, 4>, v⃗= <6, 0, 1>, and w⃗= <−1, 2, 5> \vec{u}=~<3,~-1,~4>,~\vec{v}=~<6,~0,~1>,~\text{and}~\vec{w}=~<-1,~2,~5>~lie in the same plane?
Cross Product Properties
Practice: Cross Product Properties
Let u⃗, v⃗, w⃗∈R3\vec u ,\ \vec{v} ,\ \vec w \in \reals^3 be vectors.
Let c, d∈Rc,\ d \in \reals be scalars.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 51_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$
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>} then u⃗×v⃗\bcb{\vec{u} \times \vec{v}} is parallel to:
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→\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})