If vectors v⃗=⟨3,1,0⟩v ⃗=⟨3,1,0⟩v⃗=⟨3,1,0⟩, u⃗=⟨0,−2,0⟩u ⃗=⟨0,-2,0⟩u⃗=⟨0,−2,0⟩, and w⃗=v⃗×u⃗w ⃗=v ⃗×u ⃗w⃗=v⃗×u⃗, find w⃗⃗w⃗⃗w⃗⃗.

w⃗=⟨0,0,−6⟩w⃗=\langle0,0,-6\ranglew⃗=⟨0,0,−6⟩

If vectors v ⃗ = ⟨ 3 , 1 , 0 ⟩ v ⃗=⟨3,1,0⟩ v ⃗ = ⟨ 3 , 1 , 0 ⟩ , u ⃗ = ⟨ 0 , − 2 , 0 ⟩ u ⃗=⟨0,-2,0⟩ u ⃗ = ⟨ 0 , − 2 , 0 ⟩ , and w ⃗ = v ⃗ × u ⃗ w ⃗=v ⃗×u ⃗ w ⃗ = v ⃗ × u ⃗ , find w ⃗⃗ w⃗⃗ w ⃗⃗ .

Let's see how we can solve this problem. Here we're told if vector v is equal to 3, comma, 1, comma, 0, and vector u is equal to 0, comma, negative 2, comma 0, and w is the cross product of v and u, find w. So this is a problem where we need to calculate the cross product, and in this problem, we're actually given this table that we've talked about that allows you to find the cross product between vectors. So let's go ahead and begin. Now for vector v, we're going to put the I, j, and k components that we see there in for v. So we can see that I is 3, j is 1, and k is 0. So we're going to put 310 for vector v. Now for vector u, that's going to be the I j k components of that vector. So we can see based on what we're given it's 0, negative 2, 0 that's going to be vector u. Now what I can also see here is that I need to repeat the I and j column. So I can see that the I column is 3 and then 0, because we have the vector v and the vector u, and then I can see for this column for the j's, we have 1 and negative 2. So now we've gone ahead and plugged in all the numbers, which means we need to do the cross product operation from here. Now what the cross product operation is going to tell you is what the x component is, the y component is, and the z component is, or basically the I, the j, and the k. So let's go ahead and figure this out. Now I'll start here with I, and what I'm going to do is cross through here, and then cross up there. So we're going to have 1 times 0 which is 0, and then we'll have negative 2 times 0 which is 0, and then that's going to be 0 minus 0. Now let's go to the y component which is j. If I go here through j and cross there and there, we'll have 0 times 0 which is 0 minus 0 times 3 which is 0. Now what I'll do is I'll last go to the k component, I'll cross through here and I'll cross up there. So we'll have 3 times negative 2, and 3 times negative 2 is negative 6, and that's going to be minus 0 times 1, which is 0. So this is what we end up having for the I component, for the j component, and for the k component of our vector, because recall that the cross product gives you another vector rather than a scalar. So what we can do is complete these operations. So we'll have 0 minus 0 which is 0 I, we'll have 0 minus 0 which is 0 j, and we'll have negative 6 minus 0 which is negative 6 k. So what our vector ends up becoming if we do this in the bracket notation, is we'll have that vector w is equal to 0, comma, 0, comma, negative 6, and this right here is the vector we end up getting, and that is the solution to this problem. Now I do want you to notice something interesting about the result that we got. Notice how the first vector we had had an I and j component, The second vector we had had just a j component, but the third vector we got the vector that came out with this cross product was a k component, and it's just a k component. Well, this actually makes sense, but if you think about it, the cross product gives you a vector that's perpendicular to the other 2. So since both of these were only existing in either I or j, it makes sense that we got a vector that only had a k component. Now don't worry, that's not something you need to memorize per se, but it's just something that's interesting about the result that we got, and it's something you should recognize when dealing with cross products. So hope you found this video helpful. Thanks for watching.

If vectors v⃗=⟨3,1,0⟩v ⃗=⟨3,1,0⟩v⃗=⟨3,1,0⟩, u⃗=⟨0,−2,0⟩u ⃗=⟨0,-2,0⟩u⃗=⟨0,−2,0⟩, and w⃗=v⃗×u⃗w ⃗=v ⃗×u ⃗w⃗=v⃗×u⃗, find w⃗⃗w⃗⃗w⃗⃗.

w⃗=⟨0,0,−6⟩w⃗=\langle0,0,-6\ranglew⃗=⟨0,0,−6⟩

w⃗=⟨0,−2,0⟩w⃗=\langle0,-2,0\ranglew⃗=⟨0,−2,0⟩

w⃗=⟨0,0,6⟩w⃗=\langle0,0,6\ranglew⃗=⟨0,0,6⟩

w⃗=⟨0,0,−2⟩w⃗=\langle0,0,-2\ranglew⃗=⟨0,0,−2⟩

Indice

Skip topic navigation

0. Fundamental Concepts of Algebra3h 32m

Algebraic Expressions
37m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
2m

Radical Expressions
15m

Simplifying Radical Expressions
31m

Rationalize Denominator
15m

Rational Exponents
4m

Pythagorean Theorem & Basics of Triangles
17m

1. Equations and Inequalities3h 27m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
42m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
19m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
22m

2. Graphs1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions & Graphs2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

4. Polynomial Functions1h 54m

Quadratic Functions
49m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

6. Exponential and Logarithmic Functions2h 28m

Introduction to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Introduction to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential and Logarithmic Equations
35m

7. Measuring Angles40m

Angles in Standard Position
12m

Coterminal Angles
8m

Complementary and Supplementary Angles
10m

Radians
8m

8. Trigonometric Functions on Right Triangles2h 5m

Trigonometric Functions on Right Triangles
45m

Special Right Triangles
30m

Cofunctions of Complementary Angles
26m

Solving Right Triangles
23m

9. Unit Circle1h 19m

Defining the Unit Circle
14m

Trigonometric Functions on the Unit Circle
9m

Common Values of Sine, Cosine, & Tangent
11m

Reference Angles
38m

Reciprocal Trigonometric Functions on the Unit Circle
6m

10. Graphing Trigonometric Functions1h 19m

Graphs of the Sine and Cosine Functions
32m

Phase Shifts
14m

Graphs of Secant and Cosecant Functions
10m

Graphs of Tangent and Cotangent Functions
21m

11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m

Inverse Sine, Cosine, & Tangent
28m

Evaluate Composite Trig Functions
44m

Linear Trigonometric Equations
28m

12. Trigonometric Identities 2h 34m

Introduction to Trigonometric Identities
1h 4m

Sum and Difference Identities
1h 3m

Double Angle Identities
16m

Solving Trigonometric Equations Using Identities
9m

13. Non-Right Triangles1h 38m

Law of Sines
49m

Law of Cosines
30m

Area of SAS & ASA Triangles
19m

14. Vectors2h 25m

Geometric Vectors
28m

Vectors in Component Form
32m

Direction of a Vector
23m

Unit Vectors and i & j Notation
27m

Dot Product
22m

Cross Product
11m

15. Polar Equations2h 5m

Polar Coordinate System
29m

Convert Points Between Polar and Rectangular Coordinates
26m

Convert Equations Between Polar and Rectangular Forms
29m

Graphing Other Common Polar Equations
39m

16. Parametric Equations1h 6m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

Writing Parametric Equations
21m

17. Graphing Complex Numbers1h 7m

Graphing Complex Numbers
6m

Polar Form of Complex Numbers
22m

Products and Quotients of Complex Numbers
14m

Powers of Complex Numbers (DeMoivre's Theorem)
23m

18. Systems of Equations and Matrices3h 6m

Two Variable Systems of Linear Equations
8m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

19. Conic Sections2h 36m

Introduction to Conic Sections
6m

Circles
26m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

20. Sequences, Series & Induction1h 15m

Sequences
33m

Arithmetic Sequences
25m

Geometric Sequences
15m

21. Combinatorics and Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

22. Limits & Continuity1h 49m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
27m

23. Intro to Derivatives & Area Under the Curve2h 9m

Tangent Lines & Derivatives
44m

Limits at Infinity
18m

Limits of Sequences
9m

Area Under a Curve
56m

14. Vectors

Cross Product

Precalculus

14. Vectors

Cross Product

Problema precedenteProssimo problema

Difficoltà con Precalculus?

Unisciti a migliaia di studenti che si affidano a noi per superare al meglio i loro esami!

Scelta multipla

If vectors $v ⃗=⟨3,1,0⟩$ , $u ⃗=⟨0,-2,0⟩$ , and $w ⃗=v ⃗×u ⃗$ , find $w⃗⃗$ .

$w⃗=$\langle$0,0,-6$\rangle$$

$w⃗=$\langle$0,-2,0$\rangle$$

$w⃗=$\langle$0,0,6$\rangle$$

$w⃗=$\langle$0,0,-2$\rangle$$

0 Commenti

Problema precedenteProssimo problema

Guida verificata passo dopo passo

To find the cross product of two vectors v⃗ and u⃗, we use the determinant of a 3x3 matrix. The first row of the matrix consists of the unit vectors î, ĵ, and k̂.

The second row of the matrix consists of the components of vector v⃗, which are ⟨3, 1, 0⟩.

The third row of the matrix consists of the components of vector u⃗, which are ⟨0, -2, 0⟩.

Calculate the determinant by expanding along the first row: w⃗ = î(1*0 - 0*(-2)) - ĵ(3*0 - 0*0) + k̂(3*(-2) - 1*0).

Simplify the expression: w⃗ = î(0) - ĵ(0) + k̂(-6), resulting in w⃗ = ⟨0, 0, -6⟩.