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A parallelogram in R^3 has vertices A(-1,1,2), B(1,-2,8) and C=(3,-1,2).

Related Topics

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product and Area

3 Activities

Wize University Linear Algebra Textbook > Vectors

Vector Properties

4 Activities

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).
Determine the coordinates of the fourth vertex.
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}.
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}
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 Vector Properties Questions:
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45
Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)} and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}, and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}.
Length / magnitude / norm. 19.1W
1. Find the vector d⃗\vd connecting the points P1(−2,−1)P_1(-2,-1) and P2(3,11)P_2(3,11).
2. Find the length of the vector d⃗\vd found, above. What does this length represent?
Vector Arithmetic
Suppose X=(x1,x2,x3)X=(x_1, x_2, x_3) and Y=(y1,y2,y3)Y=(y_1, y_2, y_3) If −XY⃗=<43,−35,103>\vec{-XY}=\left<\frac{4}{3},\frac{-3}{5}, \frac{10}{3}\right>, what is YX⃗\vec{YX}?
Practice Question 9: Vector Operations
Practice Question: Vector Operations
Let u⃗=(−9, 6)\vec{u}=\left(-9,\ 6\right) and v⃗=(1,−1)\vec{v}=\left(1,-1\right). If we start at the origin and travel along a directed line segment that is opposite to u⃗\vec{u} but has 1/3 of its length, then we travel along the translation of v⃗\vec{v} to your current position. Finally, we travel along the translation of the unit vector that points in the same direction as v⃗\vec{v}.
a.) Where do we end up?
If α∈R\alpha \in \mathbb{R} and v∈Vv \in V , where VV is a real vector space and αv=0\alpha v = 0 , then prove that either α=0\alpha = 0 or v=0v = 0 .
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
Practice: Magnitude of a Vector
Practice: Magnitude of a Vector
An aircraft's location is 200 miles from base, at a direction of 30° W of S, at an altitude of 35,000 ft. Find the distance between the aircraft and the base.
Note: 1 mile = 5280 feet
Practice Question: Vector Properties
Practice Question: Vector Properties
Given that u⃗\vec{u} and v⃗\vec{v} are vectors in R3\mathbb{R}^3, which of the following statements is/are always true?
Practice: Vector Operations
A parallelogram has sides ABAB, BCBC, CDCD and DADA
The following coordinates are known: A(2,0,1)A(2,0,1), C(6,0,0)C(6,0,0), and M(3,1,0)M(3,1,0), where MM is the midpoint between AA and BB
Find the vector BD⃗\vec{BD}