given that vectors(p+2q) and (5p-4q) are orthogonal,if vectors p and q are the unit vectors,find the dot product of vectors p and q?
we know that
(p+2q)•(5p-4q) = 0
5p•p + 10p•q - 4p•q - 8q•q = 0
since p and q are unit vectors, we have
5 + 6p•q - 8 = 0
p•q = 1/2
To find the dot product of vectors p and q, we can use the fact that two vectors are orthogonal if and only if their dot product is zero.
Given that vectors (p + 2q) and (5p - 4q) are orthogonal, we can express this condition as:
(p + 2q) • (5p - 4q) = 0
Expanding the dot product:
5p • p + 2q • p - 4q • p + 10p • q - 8q • q = 0
Note that since p and q are unit vectors, p • p = q • q = 1.
The expression becomes:
5 + 2q • p - 4q • p + 10p • q - 8 = 0
Combining like terms:
2q • p - 4q • p + 10p • q - 3 = 0
Rearranging:
2q • p - 4q • p + 10p • q = 3
Using the fact that p • q = q • p, we can rewrite the equation:
(q • p) (2 - 4) + 10p • q = 3
-2(q • p) + 10p • q = 3
Simplifying further:
-2(q • p) = 3 - 10p • q
Finally, we can isolate the dot product of p and q:
q • p = (3 - 10p • q) / -2
Therefore, the dot product of vectors p and q is given by (3 - 10p • q) / -2.
To find the dot product of vectors p and q, we can use the given information that vectors (p+2q) and (5p-4q) are orthogonal.
Two vectors are orthogonal if their dot product is zero. So, we can set up the dot product between (p+2q) and (5p-4q) equal to zero:
(p+2q) . (5p-4q) = 0
Now, let's simplify this expression:
(p . 5p) + (p . -4q) + (2q . 5p) + (2q . -4q) = 0
Using the properties of the dot product, we can rearrange and simplify further:
5(p . p) - 4(p . q) + 10(q . p) - 8(q . q) = 0
Since p and q are unit vectors, their magnitudes (lengths) are equal to 1, so (p . p) = 1 and (q . q) = 1. Also, the dot product is commutative, so (q . p) = (p . q).
Substituting these values in, we get:
5 - 4(p . q) + 10(p . q) - 8 = 0
Rearranging:
-4 + 6(p . q) = 0
6(p . q) = 4
Finally, dividing by 6:
(p . q) = 4/6
Simplifying further:
(p . q) = 2/3
So, the dot product of vectors p and q is 2/3.