qustion(1):find the product of xy if x,3/2,6/7,y are in a g.p
qustion(2):calculate the angle between the vector u=2i+2j and v=2i+2ji
qustion(3):if the center of a circle,the tangent from X touches A&B and AC is a diameter if AOB=47 calculateBOC
I do not understand the first question
I assume question 2 is a typo. If u = v then the angle is 0
u dot v = |u| |v| cos theta
u dot v = 2*2 + 2*2 = 8
|u| = sqrt(4+4) = 2 sqrt 2
|v| = 2 sqrt 2 also (you put in the right values)
8 = 8 cos theta = 8 * 1
cos zero = 1
3 is also a typo mess
Answer to question (1):
To find the product of xy in a geometric progression (g.p), we need to find the common ratio (r) first.
In a g.p, the ratio between any two consecutive terms is constant. Let's set x as the first term and y as the third term.
The second term, 3/2, is between x and y. So, we can write the equation:
3/2 = x * r
Similarly, we can write the equation for the third term:
6/7 = x * r^2
Now, we have two equations with two unknowns (x and r). We can solve these equations simultaneously to find the values of x and r.
Step 1: Solve the first equation (3/2 = x * r) for x:
x = (3/2) / r
Step 2: Substitute this value of x in the second equation (6/7 = x * r^2):
6/7 = [(3/2) / r] * r^2
Step 3: Simplify the equation by canceling out r terms:
6/7 = (3/2) * r
Step 4: Solve for r:
r = (6/7) / (3/2) = 4/7
Step 5: Substitute the value of r into the first equation (3/2 = x * r) to find x:
3/2 = x * (4/7)
Step 6: Solve for x:
x = (3/2) * (7/4) = 21/8
Now, we have found the values of x and r. To find the product of xy:
xy = (21/8) * (4/7) = 3/2
Therefore, the product of xy in the g.p is 3/2.
Answer to question (2):
To calculate the angle between two vectors, we can use the dot product formula:
u · v = |u| |v| cosθ
Given vector u = 2i + 2j and vector v = 2i + 2ji, we can calculate their dot product.
Step 1: Calculate |u|, the magnitude of vector u:
|u| = √(2^2 + 2^2) = √8
Step 2: Calculate |v|, the magnitude of vector v:
|v| = √(2^2 + 2^2) = √8
Step 3: Calculate the dot product of u and v:
u · v = (2)(2) + (2)(2) = 8
Step 4: Calculate the product of the magnitudes:
|u| |v| = √8 √8 = 8
Step 5: Substitute the values into the dot product formula:
8 = 8 cosθ
Step 6: Solve for cosθ:
cosθ = 8/8 = 1
Step 7: Take the inverse cosine (cos^-1) to find the angle θ:
θ = cos^-1(1) = 0 radians (or 0 degrees)
Therefore, the angle between the vectors u and v is 0 radians (or 0 degrees).
Answer to question (3):
To solve this problem, we need to use the relationship between the angles in a circle and its corresponding arcs.
Given that AC is a diameter and angle AOB = 47, we can find angle BOC as follows:
Step 1: Recall that an inscribed angle is half the measure of its intercepted arc.
Since AC is a diameter, angle AOC is a right angle, and angle AOB is given as 47 degrees.
Step 2: Apply the inscribed angle theorem to find the measure of angle BOC:
angle BOC = 2 * angle AOB
angle BOC = 2 * 47 = 94 degrees
Therefore, the measure of angle BOC is 94 degrees.