Vector Upper A Overscript right-arrow EndScripts has a magnitude of 75 units and points due west, while vector Upper B Overscript right-arrow EndScripts has the same magnitude and points due south. Find the magnitude and direction of (a) Upper A Overscript right-arrow EndScripts plus Upper B Overscript right-arrow EndScripts and (b) Upper A Overscript right-arrow EndScripts minus Upper B Overscript right-arrow EndScripts. Specify the directions relative to due west.
To find the magnitude and direction of the sum and difference of vectors A and B, we need to use vector addition and subtraction.
For the sum of vectors A and B (A + B):
1. Draw vector A with a magnitude of 75 units pointing due west.
2. Draw vector B with a magnitude of 75 units pointing due south.
3. Place the tail of vector B at the head of vector A.
4. Draw the resulting vector from the tail of A to the head of B. This is the sum of A and B.
5. Measure the magnitude of the resulting vector using a ruler or scale. Let's call it vector C.
To find the magnitude of vector C, we can use the Pythagorean theorem because vectors A and B are at right angles to each other. The Pythagorean theorem states that the square of the hypotenuse (C) is equal to the sum of the squares of the other two sides (A and B). Mathematically, it can be represented as:
C^2 = A^2 + B^2
In this case, since both A and B have the same magnitude of 75 units, the equation becomes:
C^2 = 75^2 + 75^2
Simplifying,
C^2 = 2 * 75^2
C = √(2 * 75^2)
Evaluate that expression to find the magnitude of vector C.
The direction of vector C can be found by measuring the angle between vector C and due west using a protractor or by using trigonometric ratios. In this case, the angle will be 45 degrees because vector C is pointing in the southwest direction, halfway between due west and due south.
Therefore, the magnitude of vector C is determined, and the direction relative to due west is 45 degrees (southwest).
For the difference of vectors A and B (A - B):
1. Draw vector A with a magnitude of 75 units pointing due west.
2. Draw vector B with a magnitude of 75 units pointing due south.
3. Place the tail of vector A at the head of vector B.
4. Draw the resulting vector from the tail of B to the head of A. This is the difference of A and B.
5. Measure the magnitude of the resulting vector using a ruler or scale. Let's call it vector D.
To find the magnitude of vector D, we can again use the Pythagorean theorem because vectors A and B are at right angles to each other. The equation for vector D is the same as the equation for vector C since both A and B have the same magnitude of 75 units. Calculate the magnitude of vector D using the same method as for vector C.
The direction of vector D can be found by measuring the angle between vector D and due west using a protractor or by using trigonometric ratios. In this case, the angle will be 45 degrees because vector D is pointing in the northwest direction, halfway between due west and due north.
Therefore, the magnitude of vector D is determined, and the direction relative to due west is 45 degrees (northwest).
To find the magnitude and direction of the sum and difference of the vectors Upper A Overscript right-arrow EndScripts and Upper B Overscript right-arrow EndScripts, we can use vector addition and subtraction.
(a) To find the magnitude of the sum of the vectors, we can use the Pythagorean theorem. The sum of two perpendicular sides of a right-angled triangle will give us the length of the hypotenuse.
Let's call the magnitude of Upper A Overscript right-arrow EndScripts as A and the magnitude of Upper B Overscript right-arrow EndScripts as B. Since both vectors have the same magnitude of 75 units, A = B = 75.
Using the Pythagorean theorem, we have:
Magnitude of Upper A Overscript right-arrow EndScripts plus Upper B Overscript right-arrow EndScripts = √(A^2 + B^2)
= √(75^2 + 75^2)
= √(2 * 75^2)
= √(2 * 5625)
= √11250
≈ 106.07 units
The magnitude of the sum of the vectors is approximately 106.07 units.
To find the direction, we can use trigonometry. Since Upper A Overscript right-arrow EndScripts points due west and Upper B Overscript right-arrow EndScripts points due south, the sum of the vectors will form a right-angled triangle.
Let's call the angle between due west and the resultant vector as θ. Using the tangent function:
tan(θ) = (opposite side) / (adjacent side)
= B / A
= 75 / 75
= 1
Taking the inverse tangent (arctan) of both sides:
θ = arctan(1)
≈ 45°
Therefore, the direction of the sum of the vectors relative to due west is 45°.
(b) To find the magnitude of the difference of the vectors, we can subtract the components of Upper B Overscript right-arrow EndScripts from the components of Upper A Overscript right-arrow EndScripts.
Magnitude of Upper A Overscript right-arrow EndScripts minus Upper B Overscript right-arrow EndScripts = √((A - B)^2)
= √((75 - 75)^2)
= √(0^2)
= 0 units
The magnitude of the difference of the vectors is 0 units.
Since the difference of the vectors is 0, its direction will not matter in this case.