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If α ∈ R and v ∈ V , where V is a real vector space and α v = 0 , then prove th…

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Wize University Linear Algebra Textbook > Vectors

Vector Properties

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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 .
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?
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).
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?
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}