Find the direction of the following vector: u ⃗ = ⟨ 5 √ 3 3 , 5 ⟩ u⃗=\langle\frac{5\surd3}{3},5\rangle u ⃗ = ⟨ 3 5√3 , 5 ⟩ .

this problem, we are asked to find the direction of the following vector u, and we are given u in component 4. Now to solve these types of problems, I generally first like to start by drawing this graph. And this small graph allows me to have a general understanding of the direction of the vector. Now this would be our x axis and our y axis, and if I wanna think about where this vector is going to end up, well, I see that I'm given vector u in component form. And looking at vector u, I can see that this is gonna be some positive number 5 times the square root of 3 over 3. And since this is the x component, that means we're going to be somewhere in the positive x direction. I can also see that we have the y component, which is positive 5, which would be somewhere up there. So our vector is ultimately going to look something like this, where this angle would tell us the direction of our vector. Notice we're in this first quadrant here, so all we need to do is calculate this angle theta. Now Now I can find this angle by using this equation that we learned about. The tangent of theta is equal to the y component divided by the x component. Now I can see that we are given both of these components. We have the y component, which is 5, and that's going to be divided by the x component, which is 5 times the square root of 3 over 3. Now what I can do here is I notice that we have a 5 here and there in the numerator denominator that will cancel. So So all we're going to end up with is that the tangent of theta is equal to 1 over the square root of 3 divided by 3. Now what I can do here is flip this fraction in the denominator and bring it to the numerator, and that's going to give me 3 over the square root of 3. And what I need to do is actually rationalize this now that I have a square root in the denominator. So if I multiply the top and bottom of this fraction by the square root of 3, and cancel the square roots there, giving me 3 times the square root of 3 divided by 3. And these threes are going to cancel, so all I'm really gonna end up with is the square root of 3. So we're going to have that the tangent of theta is equal to the square root of 3, which means that theta is going to be equal to the inverse tangent of the square root of 3. And the inverse tangent of the square root of 3 this turns out to be 60 degrees so that means that our angle theta is 60 degrees and that is the direction of our vector so that's how you can solve this problem hope you found this video helpful

Table of contents

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

Direction of a Vector

Precalculus

14. Vectors

Direction of a Vector

Previous problemNext problem

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Find the direction of the following vector: $u⃗=$\langle\]\frac{5\surd3}{3}$,5$\rangle$$ .

60°

0.030°

30°

0.010°

0 Comments

Previous problemNext problem

Verified step by step guidance

First, understand that the direction of a vector is given by the angle it makes with the positive x-axis. This angle can be found using the tangent function, which relates the components of the vector.

The vector u⃗ is given as ⟨$\frac{5\sqrt{3}$}{3}, 5⟩. Here, $\frac{5\sqrt{3}$}{3} is the x-component and 5 is the y-component.

To find the angle θ, use the formula $\tan$(θ) = $\frac{y}{x}$, where y is the y-component and x is the x-component of the vector.

Substitute the values into the formula: $\tan$(θ) = $\frac{5}{\frac{5\sqrt{3}$}{3}}.

Solve for θ by taking the arctangent of the result from the previous step: θ = $\arctan$$\left$($\frac{5}{\frac{5\sqrt{3}$}{3}}$\right$). This will give you the direction of the vector in degrees.