Example Question - coordinate system
Here are examples of questions we've helped users solve.
Graph Interpretation and Vector Components
<p>Considérons les coordonnées des points A et B sur le graphique :</p> <p>Point A : (2, 3)</p> <p>Point B : (7, 1)</p> <p>Calculons les composantes du vecteur \(\overrightarrow{AB}\) :</p> <p>\(\Delta x = x_B - x_A = 7 - 2 = 5\)</p> <p>\(\Delta y = y_B - y_A = 1 - 3 = -2\)</p> <p>Les composantes du vecteur \(\overrightarrow{AB}\) sont donc (5, -2).</p> <p>Utilisons ces composantes pour trouver les vecteurs \(\overrightarrow{u}\) et \(\overrightarrow{v}\) :</p> <p>\(\overrightarrow{u}\) + \(\overrightarrow{v}\) = \(\overrightarrow{AB}\)</p> <p>\(\overrightarrow{u}\) + \(\overrightarrow{v}\) = (5, -2)</p> <p>Selon l'énoncé, \(\overrightarrow{u}\) = (2, a) et \(\overrightarrow{v}\) = (b, -3).</p> <p>Donc (2 + b, a - 3) = (5, -2).</p> <p>Égalons les composantes correspondantes :</p> <p>2 + b = 5 => b = 3</p> <p>a - 3 = -2 => a = 1</p> <p>Par conséquent, les valeurs de a et b sont respectivement 1 et 3.</p>
Transformation of Function Graphs
<p>К сожалению, изображение не содержит достаточно информации для предоставления точного решения. Чтобы решить эту задачу, мне нужно увидеть графики функций \( f(x) \) и \( g(x) \), а также знать, как они связаны через простое преобразование.</p> <p>Для части a) общие виды преобразований включают: 1. Сдвиг по вертикали: \( f(x) + k \) 2. Сдвиг по горизонтали: \( f(x + h) \) 3. Растяжение по вертикали: \( a \cdot f(x) \) 4. Растяжение по горизонтали: \( f(b \cdot x) \) 5. Отражение по вертикали: \( -f(x) \) 6. Отражение по горизонтали: \( f(-x) \)</p> <p>Для части b), чтобы построить \( y=f(x-2)-1 \), нужно сделать следующее: 1. Сдвинуть график \( f(x) \) на 2 единицы вправо по оси x. 2. Сдвинуть результат вниз на 1 единицу по оси y.</p>
Identifying Quadrants on Cartesian Plane
<p>El problema muestra un plano cartesiano sin los ejes claramente definidos. Basándonos en el conocimiento estándar del plano cartesiano, podemos identificar los cuadrantes de la siguiente manera:</p> <p>El Primer cuadrante (I) se encuentra en la parte superior derecha, donde ambos, las coordenadas \( x \) y \( y \), son positivas.</p> <p>El Segundo cuadrante (II) se encuentra en la parte superior izquierda, donde las coordenadas \( x \) son negativas y las coordenadas \( y \) son positivas.</p> <p>El Tercer cuadrante (III) se encuentra en la parte inferior izquierda, donde tanto las coordenadas \( x \) como \( y \) son negativas.</p> <p>El Cuarto cuadrante (IV) se encuentra en la parte inferior derecha, donde las coordenadas \( x \) son positivas y las coordenadas \( y \) son negativas.</p> <p>Nota: Para asignar los cuadrantes correctamente en el dibujo dado, debemos asumir la orientación estándar de los ejes, aunque los ejes no están etiquetados.</p>
Guided Sketch Creation for Giulia's Movement
I can guide you on how to create the sketch based on the information given in the question. Here's the process in steps: 1. **Choose Coordinates**: First, pick a starting point for Giulia on your sketch. This point will also be the origin of your coordinate system (0,0). Let's say Giulia is running in the positive x-direction since she is running eastward. 2. **Plot the Initial Position**: At this origin, sketch Giulia at time 0. 3. **Calculate the First Position After 5 Seconds**: Considering her speed is 4.5 m/s, in 5 seconds, she will cover a distance of \( 4.5 \frac{m}{s} \times 5s = 22.5 m \). 4. **Plot the Position After 5 Seconds**: Move 22.5 units along the positive x-direction from the origin, and make a point to represent her position after 5 seconds. This is the point A. 5. **Calculate the Position After 2.5 More Seconds**: In the next 2.5 seconds, Giulia will cover an additional distance of \( 4.5 \frac{m}{s} \times 2.5s = 11.25 m \). 6. **Plot the Final Position**: From point A, move a further 11.25 units along the positive x-direction, and make a point to represent her position after 7.5 seconds in total. This is the point B. 7. **Draw the Path**: Draw a straight line from the origin through points A and B to represent her path. 8. **Label the Plot**: Mark the positions on the path with their respective times (5 seconds at A, 7.5 seconds at B). 9. **Show Coordinate Choice**: Indicate the direction of the axes. The x-axis should be marked in the eastward direction, and the y-axis should be perpendicular to it. That completes the sketch. You've now shown Giulia's movement along a straight-line path eastward into the sunrise, and you've accounted for the time intervals requested in the problem.
Calculating Distance Between Two Points in a Coordinate System
To find the distance between two points in a coordinate system, you can use the distance formula: Distance = √[(x2 - x1)² + (y2 - y1)²] Plugging in the coordinates for the two points (9,2) and (2,9), we get: x1 = 9, y1 = 2 x2 = 2, y2 = 9 Distance = √[(2 - 9)² + (9 - 2)²] Distance = √[(-7)² + (7)²] Distance = √[49 + 49] Distance = √[98] Now, we can round the result to the nearest tenth: Distance ≈ √[98] ≈ 9.899494937 Rounded to the nearest tenth, the distance is approximately 9.9 units.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com