FIND BY DRAWING AND BY CALCULATION THE RESULTANT OF TWO VECTORS 3UNITS AND 4UNITS INCLINED AT EACH OTHER AT A)30 B)90 C)120
Sure, let's find the resultant of the two vectors and have some fun while doing it!
A) 30 degrees:
To find the resultant, we'll need some drawing skills. Imagine two vectors, one pointing right and measuring 3 units, and the other pointing upwards and measuring 4 units. Now, let's draw a parallelogram with these vectors as adjacent sides. Connect the tail of the 3-unit vector to the head of the 4-unit vector. Voila! You've got yourself a resultant vector.
Now, to calculate the magnitude of the resultant vector, we can use the Pythagorean theorem. The resultant vector (R) is the hypotenuse of a right-angled triangle formed by the 3-unit and 4-unit vectors. So, R² = 3² + 4². Solving that, we get R = √(9 + 16) = √25 = 5 units.
B) 90 degrees:
Well, calculating the resultant for two vectors that are perpendicular to each other is easy peasy! If you draw the vectors, one going up (4 units) and the other going right (3 units), you'll notice that they form a right-angled triangle. The resultant vector, in this case, is simply the hypotenuse of that triangle. By again applying the Pythagorean theorem, R² = 3² + 4², we find R = √25 = 5 units.
C) 120 degrees:
This one's going to be fun! To draw the resultant vector for two vectors inclined at 120 degrees, imagine one vector going up (4 units), and the other vector going down and right (3 units). Connect the tail of the upward vector to the head of the downward vector, and voilà! You've got your resultant vector.
Now, let's find its magnitude. To make things easier, let's create a right-angled triangle with one of the vectors and the resultant. If we drop a perpendicular from the head of the resultant vector to the tail of the 4-unit vector, we can split the parallelogram into a right-angled triangle and a rectangle. And guess what? We already know the lengths of the sides of the right-angled triangle: 3 units and 4 units. So, using the Pythagorean theorem, R² = 4² - 3², which gives us R = √7 units.
Voila! We've found the resultants for all three cases. I hope my little tangent about drawing and calculations entertained you!
To find the resultant of two vectors, we can use both drawing and calculation methods. Let's go step-by-step for each case:
A) When the vectors are inclined at each other at 30 degrees:
1. Draw the two vectors: Start by drawing a horizontal line for the first vector of 3 units. Then, draw the second vector at an angle of 30 degrees with respect to the first vector, starting from the end of the first vector.
2. Measure the lengths of the two vectors: Measure the length of the first vector as 3 units and the length of the second vector as 4 units.
3. Calculate the horizontal and vertical components of each vector: For the first vector, the horizontal component is 3 units * cos(0 degrees), which equals 3 units, and the vertical component is 3 units * sin(0 degrees), which equals 0 units. For the second vector, the horizontal component is 4 units * cos(30 degrees), which equals 3.464 units, and the vertical component is 4 units * sin(30 degrees), which equals 2 units.
4. Add the horizontal and vertical components of the vectors: Add the horizontal components together and the vertical components together. The horizontal component is 3 units + 3.464 units, which equals 6.464 units, and the vertical component is 0 units + 2 units, which equals 2 units.
5. Calculate the magnitude and direction of the resultant vector: Using the Pythagorean theorem, the magnitude of the resultant vector is sqrt((6.464 units)^2 + (2 units)^2), which equals 6.762 units. The direction of the resultant vector can be found using the inverse tangent function, arctan(2 units / 6.464 units), which equals 16.9 degrees.
B) When the vectors are inclined at each other at 90 degrees:
1. Draw the two vectors: Start by drawing the first vector of 3 units horizontally. Then, draw the second vector of 4 units vertically, starting from the end of the first vector.
2. Calculate the horizontal and vertical components of each vector: For the first vector, the horizontal component is 3 units and the vertical component is 0 units. For the second vector, the horizontal component is 0 units and the vertical component is 4 units.
3. Add the horizontal and vertical components of the vectors: Add the horizontal components together and the vertical components together. The horizontal component is 3 units + 0 units, which equals 3 units, and the vertical component is 0 units + 4 units, which equals 4 units.
4. Calculate the magnitude and direction of the resultant vector: Using the Pythagorean theorem, the magnitude of the resultant vector is sqrt((3 units)^2 + (4 units)^2), which equals 5 units. The direction of the resultant vector can be found using the inverse tangent function, arctan(4 units / 3 units), which equals 53.1 degrees.
C) When the vectors are inclined at each other at 120 degrees:
1. Draw the two vectors: Start by drawing the first vector of 3 units horizontally. Then, draw the second vector of 4 units at an angle of 120 degrees counterclockwise with respect to the first vector, starting from the end of the first vector.
2. Calculate the horizontal and vertical components of each vector: For the first vector, the horizontal component is 3 units and the vertical component is 0 units. For the second vector, the horizontal component is -4 units * cos(60 degrees), which equals -2 units, and the vertical component is -4 units * sin(60 degrees), which equals -3.464 units.
3. Add the horizontal and vertical components of the vectors: Add the horizontal components together and the vertical components together. The horizontal component is 3 units + (-2 units), which equals 1 unit, and the vertical component is 0 units + (-3.464 units), which equals -3.464 units.
4. Calculate the magnitude and direction of the resultant vector: Using the Pythagorean theorem, the magnitude of the resultant vector is sqrt((1 unit)^2 + (-3.464 units)^2), which equals 3.606 units. The direction of the resultant vector can be found using the inverse tangent function, arctan((-3.464 units) / 1 unit), which equals -71.6 degrees.
Remember to add directional information to the magnitude value, as it indicates the direction and angle of the resultant vector.
To find the resultant of two vectors, you can use both drawing (graphical method) and calculation (vector addition).
A) When the vectors are inclined at an angle of 30 degrees:
1. Drawing Method:
a. Draw a line to represent the first vector with a length of 3 units and indicate the direction.
b. From the endpoint of the first vector, draw an arrow to represent the second vector with a length of 4 units at an angle of 30 degrees from the first vector.
c. The line connecting the initial point of the first vector to the endpoint of the second vector represents the resultant vector.
d. Measure the length of the resultant vector using a ruler.
2. Calculation Method:
a. Use the cosine rule to find the length of the resultant vector (R) using the magnitudes of the two vectors (A = 3 units and B = 4 units) and the angle between them (30 degrees):
R = √(A² + B² + 2ABcosθ)
R = √(3² + 4² + 2 * 3 * 4 * cos(30))
R = √(9 + 16 + 24)
R ≈ √49
R ≈ 7 units
B) When the vectors are inclined at an angle of 90 degrees:
1. Drawing Method:
a. Draw a line to represent the first vector with a length of 3 units and indicate the direction.
b. From the endpoint of the first vector, draw an arrow to represent the second vector with a length of 4 units at a right angle (90 degrees) to the first vector.
c. The diagonal connecting the initial point of the first vector to the endpoint of the second vector represents the resultant vector.
d. Measure the length of the resultant vector using a ruler.
2. Calculation Method:
a. Use the Pythagorean theorem to find the length of the resultant vector (R) using the magnitudes of the two vectors (A = 3 units and B = 4 units) since they are at a right angle:
R = √(A² + B²)
R = √(3² + 4²)
R = √(9 + 16)
R ≈ √25
R ≈ 5 units
C) When the vectors are inclined at an angle of 120 degrees:
1. Drawing Method:
a. Draw a line to represent the first vector with a length of 3 units and indicate the direction.
b. From the endpoint of the first vector, draw an arrow to represent the second vector with a length of 4 units at an angle of 120 degrees in a counterclockwise direction from the first vector.
c. The line connecting the initial point of the first vector to the endpoint of the second vector represents the resultant vector.
d. Measure the length of the resultant vector using a ruler.
2. Calculation Method:
a. Use the cosine rule to find the length of the resultant vector (R) using the magnitudes of the two vectors (A = 3 units and B = 4 units) and the angle between them (120 degrees):
R = √(A² + B² + 2ABcosθ)
R = √(3² + 4² + 2 * 3 * 4 * cos(120))
R = √(9 + 16 - 24)
R ≈ √1
R ≈ 1 unit