given that vectors(p+2q) and (5p-4q) are orthogonal,if vectors p and q are the unit vectors,find the product of vectors p and q?
cross product or dot product?
To find the product of vectors p and q, we can use the dot product formula. The dot product of two vectors is calculated by taking the product of their corresponding components and summing them up.
Let's assume vector p = (p1, p2) and vector q = (q1, q2). Since p and q are unit vectors, their magnitudes are equal to 1, which gives us:
|p| = sqrt(p1^2 + p2^2) = 1
|q| = sqrt(q1^2 + q2^2) = 1
Now, let's calculate the dot product of the given vectors (p + 2q) and (5p - 4q):
(p + 2q) · (5p - 4q)
= (p1 + 2q1)(5p1 - 4q1) + (p2 + 2q2)(5p2 - 4q2)
To solve this equation, we need to use the fact that the dot product of two orthogonal vectors is zero. Therefore, we have:
(p + 2q) · (5p - 4q) = 0
Expanding the equation:
5p1^2 - 4q1^2 + 10p1q1 - 8q1q2 + 5p2^2 - 4q2^2 + 10p2q2 - 8q1q2 = 0
Simplifying the equation:
5(p1^2 + p2^2) - 4(q1^2 + q2^2) + 2(5p1q1 + 5p2q2 - 8q1q2) = 0
Using the fact that |p| = |q| = 1, the equation further simplifies to:
5 - 4 + 10(p1q1 + p2q2 - 2q1q2) = 0
1 + 10(p1q1 + p2q2 - 2q1q2) = 0
Now, we know that vectors p and q are unit vectors, so p1q1 + p2q2 = |p||q|cos(theta), where theta is the angle between p and q.
Since both p and q are unit vectors, |p| = |q| = 1, and cos(theta) = 1 (since the vectors are orthogonal).
Therefore, p1q1 + p2q2 = 1 * 1 * 1 = 1
Substituting this value back into our equation:
1 + 10(1 - 2q1q2) = 0
10 - 20q1q2 = -1
20q1q2 = 11
q1q2 = 11/20
So, the product of vectors p and q is q1q2 = 11/20.