Two vectors, V ⃗ 1 \vec{V}_1 and V ⃗ 2 \vec{V}_2 , add to a resultant V ⃗ R = V ⃗ 1 + V ⃗ 2 \vec{V}_R = \vec{V}_1 + \vec{V}_2 . Describe V ⃗ 1 \vec{V}_1 and V ⃗ 2 \vec{V}_2 if V R 2 = V 1 2 + V 2 2 V_R^2 = V_1^2 + V_2^2 .

Hey, everyone in this problem, we're asked to consider a scenario where two forces denoted as F one and F two are applied to an object resulting in a combined force. Fr that is equal to F one plus F two. We're asked to describe the forces F one and F two, if the magnitude of the resultant force squared, so Fr squared is equal to the sum of the squares of the magnitudes of the individual forces. F one squared plus F two squared, we have four answer choices. Option A F one and F two make an angle of 60 degrees with each other. Option BF one and F two are parallel to each other. Option CF one and F two are perpendicular to each other. And option D it cannot be determined. So let's write out some of the information we were given and we were told that the magnitude of the resultant force fr squared is equal to the sum of the squares of the magnitudes of the individual forces F one squared plus F two squared. Now, if that's the case, then we can see that these must form a right triangle because the equation that we've written here is just Pythagorean theory. OK. So we have some right angle triangle and we have to figure out which side is which I know in Pythagorean theorem, the side length that's all on its own. So in this case, the fr that's our hypotenuse. OK. So the hypotenuse of our triangle is the magnitude of the resultant force fr which means the other two sides must be F one NF two. OK. So the other two sides are F one and F two, the angle between them is 90 degrees because they form this right triangle. That means that F one, the vector and F two, the vector are perpendicular and they form a 90 degree angle, they're perpendicular to each other. And so the correct answer here is option C Thanks everyone for watching. I hope this video helped see you in the next one.

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3. Vectors

Vector Composition & Decomposition

Physics

3. Vectors

Vector Composition & Decomposition

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2:26 minutes

Problem 17b

Textbook Question

Two vectors, $V ⃗ sub 1$ and $V ⃗ sub 2$ , add to a resultant ${\vec{V}}_{R} = {\vec{V}}_{1} + {\vec{V}}_{2}$ . Describe $V ⃗ sub 1$ and $V ⃗ sub 2$ if $V_{R}^{2} = V_{1}^{2} + V_{2}^{2}$ .

Verified step by step guidance

Step 1: Recognize that the equation Vᵣ² = V₁² + V₂² resembles the Pythagorean theorem, which applies to right triangles. This suggests that the vectors V₁ and V₂ are perpendicular to each other.

Step 2: Understand that when two vectors are perpendicular, their resultant vector forms the hypotenuse of a right triangle, with the two vectors as the legs of the triangle.

Step 3: To confirm this, recall that the magnitude of the resultant vector Vᵣ is given by |Vᵣ| = √(V₁² + V₂²) when the vectors are perpendicular. Squaring both sides gives Vᵣ² = V₁² + V₂², which matches the given condition.

Step 4: Conclude that V₁ and V₂ must be at a 90° angle to each other for the given relationship Vᵣ² = V₁² + V₂² to hold true.

Step 5: If needed, represent the vectors graphically or mathematically to verify their perpendicularity, ensuring that the angle between them is indeed 90°.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. This involves adding the corresponding components of the vectors, which can be visualized graphically using the head-to-tail method. The resultant vector represents the cumulative effect of the individual vectors in both magnitude and direction.

Pythagorean Theorem in Vector Context

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the context of vectors, if two vectors are perpendicular, the magnitude of the resultant vector can be calculated using this theorem, leading to the equation Vᵣ² = V₁² + V₂².

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size, regardless of its direction. It is calculated using the square root of the sum of the squares of its components. Understanding the magnitude is crucial for vector addition, especially when applying the Pythagorean theorem to find the resultant vector's length when vectors are orthogonal.

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Draw each of the following vectors. Then find its x- and y-components. F = (50.0 N, 36.9 degrees counterclockwise from the positive y-axis)

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(III) You are given a vector in the xy plane that has a magnitude of 95.0 units and a y component of -60.0 units. (a) What are the two possibilities for its 𝓍 component? (b) Assuming the 𝓍 component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the - 𝓍 direction.

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Textbook Question

At t = 0, a particle starts from rest at 𝓍 = 0, y = 0 and moves in the xy plane with an acceleration $\vec{a} = (4.0 \hat{i} + 3.0 \hat{j}) {m/s}^{2}$ . Determine the 𝓍 and y components of velocity.

\overset{⇀}{E}

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Multiple Choice

A toy gun shoots a plastic ball

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Textbook Question

Vector A is 2.80 cm long and is 60.0° above the x-axis in the first quadrant. Vector B is 1.90 cm long and is 60.0° below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of A - B In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.