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Additional practice problems--vector products

Related Topics

Wize University Linear Algebra Textbook > Products of Vectors

Dot Product Properties

5 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Volume of a Parallelepiped (Triple Product)

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product and Area

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product Properties

5 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Angle Between Vectors

6 Activities

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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}

b) u⃗ ×v⃗\vec{u}\ \times\vec{v}

2. Find cos⁡θ\cos\theta where θ\theta is the acute angle between the vectors (−4,2,0,−2)\left(-4,2,0,-2\right) and (6,−3,0,3)\left(6,-3,0,3\right).

3. Given that u⃗=(1, k+1)\vec{u}=\left(1,\ k+1\right), v⃗=(−k, 2)\vec{v}=\left(-k,\ 2\right), and θ\theta is the acute angle between the vectors, find all values of kk such that cos⁡θ=0\cos\theta=0.

4. If u⃗ ∙ v⃗=5\vec{u}\ \bullet\ \vec{v}=5 and u⃗=12v⃗\vec{u}=\frac{1}{2}\vec{v}, determine ∣∣v⃗∣∣\left|\left|\vec{v}\right|\right|.

5. Find all possible values of kk such that (1, 2k, −3, 0)\left(1,\ 2k,\ -3,\ 0\right) and (−5, 1, −k, 17)\left(-5,\ 1,\ -k,\ 17\right) are perpendicular/orthogonal.

6. If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, which of the following are defined?
i.) ∣∣u⃗+3w⃗∣∣ ∙ w⃗\left|\left|\vec{u}+3\vec{w}\right|\right|\ \bullet\ \vec{w}
ii.) u⃗×v⃗×(−w⃗)\vec{u}\times\vec{v}\times\left(-\vec{w}\right)
iii.) u⃗ ∙ (v⃗×w⃗)\vec{u}\ \bullet\ \left(\vec{v}\times\vec{w}\right)
iv.) (u⃗ ∙ v⃗)×w⃗\left(\vec{u}\ \bullet\ \vec{v}\right)\times\vec{w}

7. Find the area of the parallelogram defined by the vectors u⃗=(1,2,−1)\vec{u}=\left(1,2,-1\right) and v⃗=(0,1,2)\vec{v}=\left(0,1,2\right).

8. Find the volume of the parallelepiped defined by the vectors u⃗=(0,1,−3)\vec{u}=\left(0,1,-3\right), v⃗=(−2,5,−1)\vec{v}=\left(-2,5,-1\right) and w⃗=(−1,0,2)\vec{w}=\left(-1,0,2\right).

9. Find the area of the triangle with vertices A(1,−2,0),  B(3,0,1),  C(4,2,1)A\left(1,-2,0\right),\ \ B\left(3,0,1\right),\ \ C\left(4,2,1\right).
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})}.
133 - FML 3 - 18.1W e.g. 22
How do you express the idea that two vectors v⃗\bcb{\vec{v}} and w⃗\bcb{\vec{w}} are perpendicular, mathematically?
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.
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.
More Volume of a Parallelepiped (Triple Product) Questions:
Practice Question: Cross Product and Volume
Practice Question: Application of Cross Product
Find the volume of the parallelepiped defined by the vectors u⃗=(1,2,0), v⃗=(0,3,1)\vec{u}=\left(1,2,0\right),\ \vec{v}=\left(0,3,1\right) and w⃗=(1,1,5)\vec{w}=\left(1,1,5\right).
133 - FML 3 - 18.1W e.g. 39.2
Consider the points P1=(1,4,1)\bcb{P_1 = (1,4,1)}, P2=(3,4,−1)\bcb{P_2 = (3,4,-1)}, P3=(2,4,−2)\bcb{P_3 = (2,4,-2)}, P4=(0,2,0)\bcb{P_4 = (0,2,0)}.
Find the volume of the parallelepiped defined by the vectors P1P2⃗\bcb{\vec{P_1P_2}}, P1P3⃗\bcb{\vec{P_1P_3}} and P1P4⃗\bcb{\vec{P_1P_4}}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 39.2_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$
Consider the points P1=(1,4,1)\bcb{P_1 = (1,4,1)}, P2=(3,4,−1)\bcb{P_2 = (3,4,-1)}, P3=(2,4,−2)\bcb{P_3 = (2,4,-2)}, P4=(0,2,0)\bcb{P_4 = (0,2,0)}.
Find the volume of the parallelepiped defined by the vectors P1P2⃗\bcb{\vec{P_1P_2}}, P1P3⃗\bcb{\vec{P_1P_3}} and P1P4⃗\bcb{\vec{P_1P_4}}.
Practice Question: Cross Product and Volume
Practice Question: Application of Cross Product
Find the volume of the parallelepiped defined by the vectors u⃗=(1,2,0), v⃗=(0,3,1)\vec{u}=\left(1,2,0\right),\ \vec{v}=\left(0,3,1\right) and w⃗=(1,1,5)\vec{w}=\left(1,1,5\right).
Find the volume of the parallelepiped determined by 𝑢⃗=(1,3,3),𝑣⃗=(1,0,2),𝑢⃗ = (1,3,3), \vec𝑣 = (1,0,2), and 𝑤⃗=(0,2,3). 𝑤⃗= (0,2,3).
Find the volume of the parallelepiped defined by the vectors u→=(1,0,1)\overrightarrow{u}=\left(1,0,1\right), v→=(2,−1,0)\overrightarrow{v}=\left(2,-1,0\right)and w→=(3,−3,4)\overrightarrow{w}=\left(3,-3,4\right).
More Cross Product and Area Questions:
Practice: Cross Product and Areas
Find the area of the triangle with vertices M(−3,−1,0)M(-3,-1,0), N(3,1,−1)N(3,1,-1), and P(−2,−1,0)P(-2,-1,0)
Practice Question: Application of Cross Product
Practice Question: Application of Cross Product
Find the area of the triangle that has edges defined by 𝑢⃗=(−1,0,2)𝑢⃗ = (−1,0,2) and 𝑣=(3,−3,0)𝑣=(3,−3,0)
133 - FML 3 - 18.1W e.g. 28
Find the area of the parallelogram with vertices at (2,−3), (0,2), (13,−14),\bcb{(2,-3),~(0,2),~(13,-14),} and (11,−9)\bcb{(11,-9)}.
133 - FML 3 - 18.1W e.g. 27
Find the area of the triangle defined by the vectors [57]\bcb{\boldsymbol{ \begin{bmatrix} 5 \\ 7 \end{bmatrix}}} and [31]\bcb{\boldsymbol{ \begin{bmatrix} 3 \\ 1 \end{bmatrix}}}.
133 - FML 3 - 18.1W e.g. 39.1
Consider the points P1=(1,4,1)\bcb{P_1 = (1,4,1)}, P2=(3,4,−1)\bcb{P_2 = (3,4,-1)}, P3=(2,4,−2)\bcb{P_3 = (2,4,-2)}, P4=(0,2,0)\bcb{P_4 = (0,2,0)}.
Find the area of the triangle with vertices P1, P2, P3\bcb{P_1,\, P_2,\, P_3}.
Practice: Cross Product (2)
Let X=(0, 0, 0), Y=(1, 1, 0), & Z=(0, 1, 1).X=(0,~0,~0),~Y=(1,~1,~0),~\&~Z=(0,~1,~1)..
What is the area of the triangle XYZ?
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 39.1_$\tkcth{Mock F}\tkct{?}$_$\tkct{no vid / soln }$
Consider the points P1=(1,4,1)\bcb{P_1 = (1,4,1)}, P2=(3,4,−1)\bcb{P_2 = (3,4,-1)}, P3=(2,4,−2)\bcb{P_3 = (2,4,-2)}, P4=(0,2,0)\bcb{P_4 = (0,2,0)}.
Find the area of the triangle with vertices P1, P2, P3\bcb{P_1,\, P_2,\, P_3}.
A parallelogram in R3R^3 has vertices A(−1,1,2)A\left(-1,1,2\right), B(1,−2,8)B\left(1,-2,8\right) and C=(3,−1,2)C=\left(3,-1,2\right).
Find the area of the parallelogram defined by the vectors u→=(1,3,0)\overrightarrow{u}=\left(1,3,0\right)and v→=(−2,0,1)\overrightarrow{v}=\left(-2,0,1\right).
Cross Product and Area
Find the area of the parallelogram ABCD that has vertices A(1,2,5)A(1,2,5), B(−3,0,2)B(-3,0,2), C(−3,−5,−2)C(-3,-5,-2) and D(1,−3,1)D(1,-3,1).
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 Question: Vector Properties
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?)
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:
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})
More Angle Between Vectors Questions:
133 - FML 3 - 18.1W e.g. 47
Given u⃗=<1,0,1>\bcb{\vec{u} = \left< 1, 0, 1\right>} and v⃗=<1,0,0>\bcb{\vec{v} = \left< 1, 0, 0\right>}, compute the acute angle between u⃗\bcb{\vec{u}} and v⃗\bcb{\vec{v}}.
Consider three vectors A⃗=2i^+j^,  B⃗=i^−2j^\vec{A}=2\hat{i}+\hat{j},\,\,\vec{B}=\hat{i}-2\hat{j}, and C⃗=−2i^+j^\vec{C}=-2\hat{i}+\hat{j}. The angle between C⃗\vec{C} and A⃗×B⃗\vec{A}\times\vec{B} is?
Find the value of cos 𝜃 where 𝜃 is the acute angle between 𝑢⃗=(1,−2)𝑢⃗=(1,−2) and 𝑣⃗=(3,1).\vec𝑣 = (3,1).