The three vectors have magnitude a=3m ,b=4m and c=10m,and angle thetha =30 with positive X axis for b and c makes 90degree angle with b and a is at 0 degree with positive X axis ,if c=pa +qb ,find the value of p and q.
separating the i and j components,
a = 3i + 0j
b = 2√3 i + 2j
c = -5i + 5√3 j
So, since c = pq + qb,
3p + 2√3 q = -5
0p + 2q = 5√3
Now just solve for p and q.
I want this solution step by step then it could be easy to understand.
To find the value of p and q in the equation c = pa + qb, we need to determine the components of vectors a and b.
Let's start by finding the components of vector a:
Magnitude of a = 3 m
Angle of a with the positive X-axis = 0 degrees
Using the magnitude and angle, we can find the components of a as follows:
ax = 3 cos(0) = 3
ay = 3 sin(0) = 0
So, the components of vector a are ax = 3 and ay = 0.
Now, let's find the components of vector b:
Magnitude of b = 4 m
Angle of b with the positive X-axis = 30 degrees
Using the magnitude and angle, we can find the components of b as follows:
bx = 4 cos(30) = 4 * √3 / 2 = 2√3
by = 4 sin(30) = 4 * 0.5 = 2
So, the components of vector b are bx = 2√3 and by = 2.
We are given that vector c is a linear combination of vectors a and b, i.e., c = pa + qb. And it is also given that c makes a 90-degree angle with b.
Since vectors a and b are at right angles, their dot product is zero: a · b = 0.
Using the components of vectors a, b, and c, we can set up the equation:
(cx, cy) = p (ax, ay) + q (bx, by)
Substituting the values:
(10, 0) = p (3, 0) + q (2√3, 2)
Now we equate the x-components and y-components separately:
10 = 3p + 2√3q (Equation 1)
0 = 0p + 2q (Equation 2)
From Equation 2, we get:
2q = 0
Thus, q = 0.
Substituting q = 0 in Equation 1, we get:
10 = 3p
Solving for p:
p = 10 / 3
Therefore, the values of p and q are p = 10/3 and q = 0, respectively.
To find the values of p and q in the equation c = pa + qb, we need to break down the vector c into its components along the vectors a and b.
Let's start with the given information about the magnitudes and angles of the vectors:
Magnitude of vector a = 3m
Magnitude of vector b = 4m
Magnitude of vector c = 10m
Angle between vector b and the positive X-axis = 30 degrees
Angle between vector c and vector b = 90 degrees
Angle between vector a and the positive X-axis = 0 degrees
Step 1: Finding the X-component of vector b
To find the X-component of vector b, we can use the formula:
X-component = magnitude * cosine(angle)
X-component of vector b = 4m * cos(30 degrees)
Using the cosine value of 30 degrees (which is √3/2), we can calculate:
X-component of vector b = 4m * √3/2 = 2√3 m
Step 2: Finding the Y-component of vector c
Since the angle between vector c and vector b is 90 degrees, the Y-component of vector c must be the magnitude of vector c itself.
Y-component of vector c = magnitude of vector c = 10m
Step 3: Finding the Y-component of vector b
To find the Y-component of vector b, we can use the formula:
Y-component = magnitude * sine(angle)
Y-component of vector b = 4m * sin(30 degrees)
Using the sine value of 30 degrees (which is 1/2), we can calculate:
Y-component of vector b = 4m * 1/2 = 2m
Step 4: Expressing vector c in terms of vectors a and b
Now that we have the X and Y components of vectors b and c, we can express vector c in the form pa + qb.
The X-component of vector c is equal to p times the X-component of vector a plus q times the X-component of vector b:
2√3 m = p * (3m * cos(0 degrees)) + q * (2√3 m)
Simplifying the equation:
2√3 m = 3m * p + 2√3 m * q
Comparing the coefficients of √3:
2 = 2q
Thus, q = 1.
Step 5: Finding the value of p
Substituting the value of q into the equation from Step 4, we have:
2√3 m = 3m * p + 2√3 m * 1
Since the coefficients of √3 on both sides are equal, we can say:
3m * p = 0
To solve for p, we divide both sides by 3m:
p = 0
Therefore, the value of p is 0 and the value of q is 1.