How to use vectors to solve a word problem

Brian McLogan
12 Mar 201909:57
EducationalLearning
32 Likes 10 Comments

TLDRThe video script presents a detailed explanation of vector addition, specifically focusing on the scenario of a boat traveling against a current. The speaker begins by drawing two vectors representing the boat's northward movement at 27 miles per hour and a south 60 degrees west flowing current at 8 miles per hour. Instead of using the component form, which would require calculating x and y coordinates, the speaker opts for a more straightforward approach by expressing each vector in terms of its magnitude and direction. The boat vector is represented as 27 miles per hour at a 90-degree angle (due north), while the current vector is 8 miles per hour at a 210-degree angle. The resultant vector, found by adding the vectors head-to-tail, gives the actual speed and direction of the boat's movement. The magnitude of the resultant vector is calculated using the Pythagorean theorem, resulting in approximately 24 miles per hour. The direction is found using the arctangent function, yielding a negative 73-degree angle, which is then converted to a bearing of north 17 degrees west. The explanation emphasizes the practical application of vector addition in real-world scenarios, such as navigating a boat against a current.

Takeaways
  • 🚤 The main subject is the vector representation of a boat's movement and the effect of a current on its trajectory.
  • 🧭 The boat is traveling due north at a speed of 27 miles per hour, which is the magnitude of the vector representing the boat.
  • 🔄 There's a current flowing at a bearing of south 60 degrees west with a speed of 8 miles per hour, which is the magnitude of the current vector.
  • 📐 The direction of the boat vector is 90 degrees, while the current vector's direction is 210 degrees in standard form.
  • 🤔 The challenge is to find the actual speed and direction of the boat when affected by the current, which involves adding the two vectors.
  • 📊 Vector addition is done by adding the corresponding components of the vectors, which in this case are the cosines and sines of their respective angles.
  • 🔢 The magnitude of the resultant vector (b + c) is found by taking the square root of the sum of the squares of its components.
  • 🧮 The direction (angle) of the resultant vector is found using the arctangent function, considering the signs of the components.
  • 📉 The magnitude of the resultant vector is expected to be lower than the boat's original speed due to the opposing force of the current.
  • 🗺 The direction of the resultant vector is shifted from the original northward direction of the boat due to the influence of the current.
  • 🔄 Understanding the concept of vector addition and its application to real-world scenarios, such as boat navigation, is crucial for solving such problems.
Q & A

  • What is the magnitude of the boat's vector?

    -The magnitude of the boat's vector is 27 miles per hour, as it is traveling due north at that rate.

  • What is the bearing direction of the current?

    -The bearing direction of the current is south 60 degrees west.

  • What is the magnitude of the current's vector?

    -The magnitude of the current's vector is 8 miles per hour.

  • Why is it easier to use the magnitude and direction of vectors rather than their component form in this context?

    -Using the magnitude and direction is easier because it avoids the need to calculate the x and y coordinates of the terminal points, which would require additional trigonometric calculations.

  • How is the boat's vector represented in terms of its magnitude and direction?

    -The boat's vector is represented as 'b' for boat, with a magnitude of 27 and an angle of 90 degrees, using cosine and sine functions in standard form.

  • How is the current's vector represented in terms of its magnitude and direction?

    -The current's vector is represented as 'c' for current, with a magnitude of 8 and an angle of 210 degrees, again using cosine and sine functions in standard form.

  • What is the method used to find the actual speed and direction of the boat?

    -The actual speed and direction of the boat are found by summing the two vectors (boat and current) using the head-to-tail method and calculating the resultant vector.

  • How are the components of the resultant vector calculated?

    -The components of the resultant vector are calculated by adding the corresponding components of the boat and current vectors: (27*cos(90) + 8*cos(210)), (27*sin(90) + 8*sin(210)).

  • What is the magnitude of the resultant vector representing the actual speed of the boat?

    -The magnitude of the resultant vector is approximately 24 miles per hour, after rounding the components and calculating the square root of their squares' sum.

  • What is the direction of the resultant vector?

    -The direction of the resultant vector is found using the tangent function and is approximately -73 degrees, which translates to a bearing of North 17 degrees West.

  • Why is it important to consider the direction of the resultant vector in the context of the problem?

    -The direction of the resultant vector is important because it shows the actual path the boat is taking when influenced by the current, which is crucial for navigation and correcting the course if necessary.

  • What is the significance of the boat's vector being shifted to the left due to the current?

    -The shift to the left indicates that the current is affecting the boat's northward travel, causing it to drift towards the west, which is essential for understanding the boat's true motion and planning the navigation accordingly.

Outlines
00:00
🚢 Vector Analysis of a Boat's Movement

The first paragraph discusses the vector representation of a boat's movement. The speaker explains that using the component form is not very helpful and instead opts to draw the vector directly. A vector 'b' for the boat is created, representing its northward movement at a speed of 27 miles per hour. Additionally, a current vector 'c' is introduced, flowing at a bearing of south 60 degrees west at 8 miles per hour. The speaker emphasizes the need to understand the resultant vector, which is the combined effect of the boat's movement and the current, to determine the actual speed and direction of the boat. They also mention that using trigonometric functions to find the x and y coordinates is possible but more complex. Instead, the speaker suggests writing each vector in terms of its magnitude and direction, which simplifies the calculation process.

05:01
🔢 Calculating the Resultant Vector

The second paragraph focuses on the mathematical process of adding two vectors to find the resultant vector. The speaker clarifies that to add vectors, one must add their respective components. The resultant vector is calculated by adding the cosine and sine components of the boat's vector 'b' and the current's vector 'c'. The speaker uses a calculator to find the components of the resultant vector, storing intermediate results for accuracy. The magnitude of the resultant vector is then found using the Pythagorean theorem, and the direction (angle) is determined using the arctangent function. The speaker rounds the final direction to -73 degrees, which is then converted to a bearing of north 17 degrees west. The paragraph concludes with a confirmation that the magnitude of the resultant vector is lower than the boat's original speed due to the impact of the current, and the vector is shifted to the left, aligning with the expected outcome.

Mindmap
Keywords
💡Vector
A vector is a mathematical concept that has both magnitude (size) and direction. In the video, vectors are used to represent the movement of a boat and the current it is facing. The boat's vector is due north with a magnitude of 27 miles per hour, while the current's vector is south 60 degrees west with a magnitude of 8 miles per hour. Vectors are essential for understanding how the boat's actual speed and direction are affected by the current.
💡Bearing
Bearing refers to the direction of travel or the position of an object with reference to another. In the script, the boat's bearing is 'due north,' and the current's bearing is 'south 60 degrees west.' These bearings are used to define the direction components of the vectors representing the boat and the current.
💡Magnitude
The magnitude of a vector is its length, which corresponds to the speed or rate of movement in the context of the video. The boat's magnitude is given as 27 miles per hour, and the current's magnitude is 8 miles per hour. Magnitude is a critical component in calculating the resultant vector that determines the actual speed and direction of the boat.
💡Component Form
Component form is a way of representing a vector by breaking it down into its orthogonal (x and y) components. The video discusses the challenge of using component form without knowing the x and y coordinates of the terminal points, which is why the video opts for representing vectors by their magnitude and direction instead.
💡Trigonometry
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In the video, trigonometric functions (cosine and sine) are used to express the vectors in terms of their magnitude and direction. For example, the boat's vector is represented as 27 times the cosine of 90 degrees and the sine of 90 degrees.
💡Resultant Vector
The resultant vector is the vector sum of two or more vectors. In the context of the video, the resultant vector is found by adding the boat's vector and the current's vector. This gives the actual speed and direction of the boat when the effect of the current is considered.
💡Calculator
A calculator is an electronic device used for performing mathematical calculations. In the script, the presenter mentions using a calculator to find the magnitude and direction of the resultant vector by performing operations such as squaring, square root, and tangent inverse on the stored values of the vector components.
💡Standard Form
Standard form refers to the conventional way of expressing angles and vectors that a calculator can understand. The video emphasizes using standard form (as opposed to bearings) because calculators are not designed to interpret bearings directly. For instance, the current's angle is converted from a bearing to 210 degrees in standard form for calculation purposes.
💡Direction
Direction refers to the path or orientation of movement. The video aims to find the actual direction of the boat, which is influenced by the current. The direction is determined by calculating the angle of the resultant vector, which is found to be 17 degrees west of north after adjusting for the quadrant.
💡Speed
Speed is the rate at which an object moves, typically measured in terms of distance over time. The video calculates the actual speed of the boat by finding the magnitude of the resultant vector after considering the boat's speed and the impact of the current. The actual speed is found to be 24 miles per hour.
💡Quadrant
In the context of the video, a quadrant refers to one of the four sections created by the intersection of the x and y axes in the Cartesian coordinate system. The direction of the resultant vector is initially found in the second quadrant, but the video adjusts this to the fourth quadrant to determine the bearing of the boat's actual direction.
Highlights

The speaker introduces a vector representing a boat traveling due north at 27 miles per hour.

A current vector is introduced, flowing at a bearing of south 60 degrees west at 8 miles per hour.

The magnitude of the boat's vector is determined by its speed, as no distance is given.

The current's vector is significantly smaller due to its lower speed.

The speaker opts to use the magnitude and direction of vectors instead of component form for simplicity.

The boat's vector is represented in terms of its magnitude (27) and direction (90 degrees).

The current's vector is calculated with a magnitude of 8 and a direction of 210 degrees.

The resultant vector is found by summing the boat's vector and the current's vector.

The speaker uses the head-to-tail method to visualize the resultant vector.

The magnitude of the resultant vector is calculated using the cosine and sine of the vectors' angles.

The direction of the resultant vector is found using the tangent function and the inverse tangent function.

The speaker stores intermediate results in the calculator for ease of calculation.

The magnitude of the resultant vector is approximately 24 miles per hour.

The direction of the resultant vector is found to be approximately 73 degrees, adjusted to a bearing of north 17 degrees west.

The speaker emphasizes the practicality of understanding vector addition in navigation and real-world applications.

The impact of the current on the boat's direction and speed is clearly demonstrated through vector addition.

The speaker provides a step-by-step guide on how to perform vector addition in terms of magnitude and direction.

The importance of considering the direction of vectors when calculating the resultant vector is highlighted.

The transcript concludes with a summary of the process and an encouragement to apply this method to similar problems.

Transcripts

Browse More Related Video

AP Physics B Kinematics Presentation General Problems #08

Adding Vector Components Example | Physics with Professor Matt Anderson | M3-11

Acceleration | One-dimensional motion | Physics | Khan Academy

AP Physics B Kinematics Presentation #51

AP Physics B Kinematics Presentation #48

1-2 C Vector Addition - 2 (Parallelogram Method)

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Vector AnalysisPhysicsMotionBoat JourneyCurrent ImpactSpeed CalculationDirection ChangeTrigonometryBearing AnglesResultant VectorNavigation