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Given that u, vand ware vectors in R^3and kis a scalar (constant). Which of theโ€ฆ

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Wize University Linear Algebra Textbook > Products of Vectors

Dot Product Properties

5 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product Properties

5 Activities

Given that uโ†’\overrightarrow{u}, vโ†’\overrightarrow{v}and wโ†’\overrightarrow{w}are vectors in R3\mathbb{R}^3and kkis a scalar (constant). Which of the following operations are defined (possible)?
  1. uโ†’โ‹…(vโ†’ร—wโ†’)\overrightarrow{u}\cdot(\overrightarrow{v}\times\overrightarrow{w})
  2. uโ†’+(vโ†’โ‹…wโ†’)\overrightarrow{u}+(\overrightarrow{v}\cdot\overrightarrow{w})
  3. uโ†’+(vโ†’ร—wโ†’)\overrightarrow{u}+(\overrightarrow{v}\times\overrightarrow{w})
  4. k(uโ†’โ‹…vโ†’)k(\overrightarrow{u}\cdot\overrightarrow{v})
  5. kโˆฅuโ†’โˆฅโ‹…(vโ†’ร—wโ†’)k\|\overrightarrow{u}\|\cdot(\overrightarrow{v}\times\overrightarrow{w})
More Dot Product Properties Questions:
133 - FML 3 - 18.1W e.g. 46
If โˆฃuโƒ—โˆฃ=2\bcb{|{\vec{u}|} = 2} and โˆฃvโƒ—โˆฃ=3\bcb{|{ \vec{v} |} = 3}, where uโƒ—โ‹…vโƒ—=2\bcb{\vec{u} \cdot \vec{v} = 2}, find โˆฃuโƒ—โˆ’vโƒ—โˆฃ\bcb{|{\vec{u} - \vec{v} }|}.
133 - FML 3 - 18.1W e.g. 40
If uโƒ—=<1,โˆ’2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>} and vโƒ—=<2,โ€‰โˆ’1,โ€‰3>\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}, find (2uโƒ—)โ‹…(3vโƒ—)\bcb{(2\vec{u})\cdot (3\vec{v})}.
133 - FML 3 - 18.1W e.g. 22
How do you express the idea that two vectors vโƒ—\bcb{\vec{v}} and wโƒ—\bcb{\vec{w}} are perpendicular, mathematically?
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that uโƒ—, vโƒ—, wโƒ—\vec{u},\ \vec{v},\ \vec{w} are vectors in R2R^2, pโƒ—, qโƒ—, rโƒ—\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
๐‘ขโƒ—,๐‘ฃโƒ—,๐‘คโƒ—๐‘ขโƒ— , \vec๐‘ฃ , ๐‘คโƒ— are vectors in R3โ„^3 and ๐‘, ๐‘‘ are scalars. Which of the following is/are true?
Dot Product Properties
Practice: Dot Product Properties
Let uโƒ—, vโƒ—,wโƒ—โˆˆR3\vec{u},\ \vec{v}, \vec{w} \in \reals^3 be vectors and let c, dโˆˆRc,\ 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 uโƒ—=<1,k,3>\bcb{\vec{u} = \left< 1, k, 3 \right>} is perpendicular to the vector vโƒ—=<2,โˆ’1,k>\bcb{\vec{v} = \left< 2, -1, k\right>}.
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that uโƒ—, vโƒ—, wโƒ—\vec{u},\ \vec{v},\ \vec{w} are vectors in R2R^2, pโƒ—, qโƒ—, rโƒ—\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 ๐‘˜ such that the vectors ๐‘ขโƒ—=(1,๐‘˜,2๐‘˜)๐‘ขโƒ—=(1,๐‘˜,2๐‘˜) ๐‘Ž๐‘›๐‘‘ ๐‘ฃโƒ—=(3,1,0)๐‘ฃโƒ—=(3,1,0) are orthogonal.
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If uโƒ—=(โˆ’2,3,โˆ’1),  vโƒ—=(โˆ’1,โˆ’2,3)\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right), find
a) uโƒ— โˆ™ vโƒ—\vec{u}\ \bullet\ \vec{v}
Given that uโ†’=(5,โˆ’1,2,k)\overrightarrow{u}=\left(5,-1,2,k\right)and vโ†’=(โˆ’10,2,โˆ’4,6)\overrightarrow{v}=\left(-10,2,-4,6\right), which of the following statements is/are true?
If k=3k=3, then uโ†’\overrightarrow{u}and vโ†’\overrightarrow{v}are parallel/collinear
If k=โˆ’10k=-10, then uโ†’\overrightarrow{u}and vโ†’\overrightarrow{v}are orthogonal/perpendicular
Dot Product Properties
Find all values of kk such that the vectors u=(โˆ’1,2,k,0)\bm{u}=(-1,2,k,0) and v=(k,โˆ’6,6,0)\bm{v}=(k,-6,6,0) are orthogonal.
More Cross Product Properties Questions:
Practice Question: Dot and Cross Product Properties
Practice Question: Dot and Cross Product Properties
Given that uโƒ—=(1,โˆ’2,0,1)\vec{u}=\left(1,-2,0,1\right) and ๐‘ฃโƒ—\vec๐‘ฃ =(0,1,2,1),=(0,1,2,1), which of the following is true?
๐‘ขโƒ—,๐‘ฃโƒ—,๐‘คโƒ—๐‘ขโƒ— ,\vec{๐‘ฃ} , \vec๐‘ค are vectors in R3โ„^3 and ๐‘, ๐‘‘ are scalars. Which of the following produce(s) a vector?
i) (๐‘คโƒ—ร—๐‘๐‘ขโƒ—)+๐‘‘๐‘ฃโƒ—(\vec๐‘คร— ๐‘๐‘ขโƒ— ) + ๐‘‘\vec๐‘ฃ
ii) (wโƒ—ร—๐‘๐‘ขโƒ—)โ‹…๐‘‘๐‘ฃ(\vec{w}ร—๐‘๐‘ขโƒ— ) \cdot ๐‘‘๐‘ฃ
Practice Question: Properties of Dot and Cross Products
Practice Question: Properties of Dot and Cross Products
๐‘ขโƒ—,๐‘ฃโƒ—,๐‘คโƒ—๐‘ขโƒ— ,\vec{๐‘ฃ} , \vec๐‘ค are vectors in R3โ„^3 and ๐‘, ๐‘‘ are scalars. Which of the following produce(s) a vector?
i) (๐‘คโƒ—ร—๐‘๐‘ขโƒ—)+๐‘‘๐‘ฃโƒ—(\vec๐‘คร— ๐‘๐‘ขโƒ— ) + ๐‘‘\vec๐‘ฃ
Practice Question: Vector Properties
Practice Question: Vector Properties
If uโƒ—, vโƒ—, wโƒ—  โˆˆ 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?)
Practice: Dot and Cross Product Properties
Practice: Dot and Cross Product Properties
๐‘ขโƒ—,๐‘ฃโƒ—,๐‘คโƒ—๐‘ขโƒ— , \vec๐‘ฃ , ๐‘คโƒ— are vectors in R3โ„^3 and ๐‘, ๐‘‘ are scalars. Which of the following is/are true?
Practice: Cross Product (1)
Do the vectors uโƒ—= <3, โˆ’1, 4>, vโƒ—= <6, 0, 1>, and wโƒ—= <โˆ’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 uโƒ—, vโƒ—, wโƒ—โˆˆR3\vec u ,\ \vec{v} ,\ \vec w \in \reals^3 be vectors.
Let c, dโˆˆRc,\ 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 uโƒ—=<1,โˆ’2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>} and vโƒ—=<2,โˆ’1,3>\bcb{\vec{v} = \left< 2, -1, 3 \right>} then uโƒ—ร—vโƒ—\bcb{\vec{u} \times \vec{v}} is parallel to:
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If uโƒ—=(โˆ’2,3,โˆ’1),  vโƒ—=(โˆ’1,โˆ’2,3)\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right), find
a) uโƒ— โˆ™ vโƒ—\vec{u}\ \bullet\ \vec{v}