Adição e subtração de vetores - Como calcular o vetor resultante

O Físico

6 Feb 201911:46

Summary

TLDRThis lesson covers the concepts of vector addition and subtraction, using both the polygon and parallelogram rules. The instructor demonstrates how to calculate vector sums and differences through graphical representations, starting with simple vertical and horizontal vectors, and then more complex angled vectors. The lesson also introduces the Pythagorean theorem for right-angle triangles and a formula for calculating vector sums and differences when vectors form an arbitrary angle. Practical examples involving forces and real-world applications are provided, ensuring that learners can grasp the methodology effectively.

Takeaways

📐 The lesson covers the addition and subtraction of vector quantities, focusing on rules like the polygon and parallelogram methods.
📊 To calculate the sum of vectors, the 'polygon rule' is applied by drawing the vectors consecutively and connecting the origin to the endpoint.
➡️ Vector addition example: Three vectors A, B, and C are drawn, and the resulting vector S is the sum of A + B + C.
📏 Vector subtraction is achieved by reversing the direction of one vector and then applying the same rules as addition.
📉 Each small square in the diagrams represents a distance of 1 meter, making it possible to calculate both the direction and magnitude of vectors.
📐 Using the Pythagorean theorem is crucial when working with right-angled triangles formed by vectors to calculate the magnitude of the resultant vector.
✖️ For subtraction, the procedure is similar to addition but with the second vector reversed in direction.
📏 In cases where vectors form an angle (not a right triangle), the general vector formula involves cosine of the angle between them.
🔄 The 'parallelogram rule' allows calculating the resultant vector by constructing parallel lines and drawing the diagonal as the result.
🔢 Example of vector force calculation: When forces are represented as vectors, the formula involving cosine is used to compute the magnitude of the resultant force.

Q & A

What is the primary topic of the lesson?
-The lesson focuses on vectors, specifically the addition and subtraction of vector quantities using methods like the polygon rule and the parallelogram rule.
What is the first step in performing vector addition using the polygon rule?
-The first step is to draw the first vector (A) and, once that is completed, to draw the second vector (B) starting from the endpoint of the first vector.
How is the resultant vector defined in vector addition?
-The resultant vector (S) in vector addition is the vector that starts from the origin of the first vector and ends at the endpoint of the last vector in the sequence.
How is the magnitude of the resultant vector calculated when adding vectors?
-The magnitude of the resultant vector can be calculated by summing the magnitudes of the individual vectors, assuming they are in the same direction. In other cases, methods like the Pythagorean theorem or vector addition formulas may be used.
What happens to the direction of a vector when subtracting one vector from another?
-When subtracting a vector, the direction of the subtracted vector is reversed. This means it has the same magnitude as the original vector but points in the opposite direction.
How do you calculate the magnitude of the difference between two vectors?
-To calculate the magnitude of the difference between two vectors, you subtract the magnitude of one vector from the other if they are aligned. If they form an angle, you may need to apply the Pythagorean theorem or vector subtraction formulas.
In the lesson, what mathematical rule is applied when vectors form a right triangle?
-The Pythagorean theorem is applied when vectors form a right triangle to calculate the magnitude of the resultant vector (hypotenuse).
What is the parallelogram rule for vector addition?
-In the parallelogram rule, two vectors are drawn from the same origin. Then, parallel lines are drawn from the endpoints of each vector to form a parallelogram. The resultant vector is the diagonal of the parallelogram starting from the common origin.
How do you calculate the magnitude of the resultant vector when the vectors form an angle?
-When vectors form an angle, the magnitude of the resultant vector can be calculated using the formula: R² = A² + B² + 2AB * cos(θ), where θ is the angle between the vectors.
What formula is used when subtracting vectors that form an angle?
-When subtracting vectors that form an angle, the formula used is: R² = A² + B² - 2AB * cos(θ), where θ is the angle between the vectors.
In the example given, how is the resultant force calculated when two forces of 7 N and 8 N form an angle of 60°?
-The resultant force is calculated using the formula R² = A² + B² + 2AB * cos(θ). For forces of 7 N and 8 N with a 60° angle, the formula becomes: R² = 7² + 8² + 2(7)(8) * cos(60°), leading to a resultant force of 13 N.

Outlines

00:00

🔢 Introduction to Vector Addition and Subtraction

This paragraph introduces the lesson on vectors, specifically focusing on vector addition and subtraction. The instructor starts by explaining the 'polygon rule' for vector addition using three vectors (A, B, and C) as examples. To find the sum of these vectors, the procedure involves drawing vector A first, followed by B and then C. The resultant vector (S) is drawn from the origin of vector A to the endpoint of vector C. The instructor further illustrates the concept with a few examples, demonstrating both the vector sum and the difference. For subtraction, vectors are reversed, and the difference is calculated accordingly.

05:01

📏 Using the Pythagorean Theorem for Vector Calculation

This paragraph discusses using the Pythagorean theorem to calculate the magnitude of vectors. After drawing vectors A and B, the resultant vector (X) is formed. The values of the two vectors are given, creating a right triangle. The theorem is used to find the hypotenuse (vector sum), showing step-by-step calculations: X² = 4² + 3², resulting in X = 5 meters. The paragraph also explains the subtraction process (Y) by reversing vector B and finding the resultant. It highlights that this triangle also results in a 5-meter magnitude, showing that the rule works symmetrically for both addition and subtraction.

10:03

📐 Calculating the Resultant Using the Law of Cosines

This section explores vector addition and subtraction when vectors form an angle other than 90 degrees. Since the Pythagorean theorem is no longer applicable, the instructor introduces the Law of Cosines. For the sum, the formula is: S² = A² + B² + 2AB cos(α), where α is the angle between A and B. For subtraction, a similar formula is used, but with a minus sign instead of plus. The instructor provides a detailed example using force vectors: A = 7N, B = 8N, and α = 60°. The resultant force F is calculated step-by-step using the formula, yielding a final value of 13N.

Mindmap

Keywords

💡Vector

A vector is a quantity that has both magnitude and direction. In the video, the concept of a vector is fundamental as the lesson revolves around adding and subtracting vectors. For example, vectors A, B, and C are used in the demonstration to find the resultant vector through different methods, such as the polygon rule and parallelogram rule.

💡Resultant Vector

The resultant vector is the vector that represents the combined effect of two or more vectors. In the video, the resultant vector is often the sum of vectors A, B, and C. The video shows how to calculate the resultant using geometric methods, such as starting at the origin of one vector and ending at the destination of the last vector.

💡Polygon Rule

The polygon rule is a graphical method used to add vectors. It involves placing vectors head-to-tail in sequence, and the resultant vector is drawn from the starting point to the endpoint of the final vector. The video demonstrates this method by drawing three vectors (A, B, and C) and showing how to determine the resultant vector using this rule.

💡Parallelogram Rule

The parallelogram rule is another graphical method for adding vectors, where two vectors are drawn from a common point, and the resultant vector is represented by the diagonal of the parallelogram formed. The video illustrates this concept by showing how to draw parallel lines to each vector and where the diagonals intersect to find the resultant vector.

💡Vector Addition

Vector addition is the process of combining two or more vectors to find their resultant. The video explains this by showing how to add vectors A, B, and C both graphically and through calculations. For example, the addition of vectors A and B results in a new vector X, and the sum is visually represented in diagrams.

💡Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which is equivalent to adding one vector to the negative of another. In the video, the difference between vectors A and B is represented by vector Y, where B is reversed in direction to form -B. The resultant vector shows the change from A to B.

💡Magnitude

The magnitude of a vector refers to its length or size, often representing quantities like force or displacement. The video frequently calculates the magnitude of vectors, such as when the magnitude of vector X is found to be 5 meters after using the Pythagorean theorem on a right triangle formed by two vectors.

💡Pythagorean Theorem

The Pythagorean theorem is a mathematical formula used to calculate the length of the hypotenuse of a right triangle. In the video, this theorem is applied to determine the magnitude of the resultant vector in cases where vectors form a right-angled triangle. For example, the theorem is used when adding two perpendicular vectors with lengths of 3 and 4 units, resulting in a resultant vector of 5 units.

💡Cosine Rule

The cosine rule is used to find the magnitude of the resultant vector when the angle between two vectors is not 90 degrees. The video introduces this rule in situations where the vectors form an angle, such as when vectors A and B make a 60-degree angle, and the magnitude of the resultant vector is calculated using the cosine of the angle.

💡Force

Force is a vector quantity that describes the push or pull on an object and is measured in Newtons. In the video, forces are represented by vectors, and the resultant force is calculated using vector addition. For example, a vector with a magnitude of 7 Newtons is combined with another vector of 8 Newtons, and their resultant force is found through the parallelogram rule.

Highlights

Introduction to the lesson on vectors, focusing on addition and subtraction of vector quantities.

Explanation of the polygon rule for adding vectors, using vectors A, B, and C as an example.

Step-by-step illustration of how to draw vectors A, B, and C and calculate the resultant vector sum S = A + B + C.

Method for calculating the sum of two vectors by connecting them in sequence, showing how the resultant vector starts at the origin of A and ends at the destination of C.

Clarification of vector difference by showing how to calculate vector Y = A - B and explaining the role of vector direction in subtraction.

Use of unit squares (each representing 1 meter) to calculate the magnitude of resultant vectors and explain how direction and magnitude are visualized.

Detailed explanation of vector subtraction by reversing the direction of vector B when calculating A - B.

Explanation of how to calculate the magnitude of the resultant vector using the Pythagorean theorem when vectors form a right triangle.

Further clarification of vector addition and subtraction when vectors are not aligned vertically or horizontally, emphasizing geometric interpretation.

Introduction to the rule of the parallelogram for vector addition when vectors have coincident origins.

Explanation of how to draw parallel lines to the original vectors and determine the resultant vector using the parallelogram rule.

Introduction of the cosine rule for calculating vector magnitude when vectors form an arbitrary angle, emphasizing its use over the Pythagorean theorem for non-right triangles.

Demonstration of the cosine rule in the context of vectors representing forces, with a practical example involving vectors A and B.

Computation of the magnitude of the resultant force vector using the cosine rule with a given angle of 60° between the vectors.

Final summary of the lesson with a practical example and explanation of how to apply both the polygon rule and the parallelogram rule for vector addition and subtraction.