If vector a has a magnitude of 29 units and points in the positive y direction. when vector b is added to a the resultant is 24 units in the negitive y direction, what is the magnitude of b?
Do simple vector addition:
a = (0,29)
b = (x,y)
a+b = (0,-24)
0+x = 0 ==> x = 0
29 + y = -24 ==> y = -53
So, b = (0,-53)
To find the magnitude of vector b, we can use the concept of vector addition. Since vector a points in the positive y direction and has a magnitude of 29 units, we can represent it as a = 0i + 29j, where i and j are unit vectors in the x and y directions, respectively.
The resultant vector, when vector b is added to vector a, points in the negative y direction and has a magnitude of 24 units. Let's represent vector b as b = bx * i + by * j, where bx and by are the respective x and y components of vector b.
Now, we can perform vector addition:
a + b = 0i + 29j + bx * i + by * j
= bx * i + (29 + by) * j
Since the resultant points in the negative y direction, the y component of the sum must be -24 units:
29 + by = -24
Solving for by:
by = -24 - 29
= -53
Therefore, vector b can be represented as b = bx * i - 53j.
To find the magnitude of vector b, we can use the Pythagorean theorem:
|b| = √(bx^2 + by^2)
Since vector b points only in the y direction (negative y), bx = 0. Thus, the magnitude of vector b is:
|b| = √(0^2 + (-53)^2)
= √(0 + 2809)
= √2809
= 53 units
Therefore, the magnitude of vector b is 53 units.
To find the magnitude of vector b, we need to analyze the given information.
Let's assume the magnitude of vector b is 'b units'.
From the given information, we know that:
- The magnitude of vector a is 29 units.
- Vector a points in the positive y direction.
- When vector b is added to vector a, the resultant is 24 units in the negative y direction.
Now, let's break down the vectors into their components along the y-axis:
The y-component of vector a is positive and equal to the magnitude of vector a, which is 29 units. Therefore, we have:
a𝑦 = +29 units.
The y-component of vector b is negative and unknown, which we will denote as -b. Hence, we have:
b𝑦 = -b units.
When vector a and vector b are added, the resultant has a magnitude of 24 units in the negative y direction. This means that the y-component of the resultant is negative and equal to -24 units.
Therefore, we can write the equation:
a𝑦 + b𝑦 = -24 units.
Substituting the given values, we get:
29 - b = -24.
Simplifying the equation further:
29 + 24 = b.
Thus, the magnitude of vector b is 53 units.