if a & b are perpendicular vectors, show that
a)(a+2b).(a-b)=|a|^2-2|b|^2
b)(a+b).(a-b)=|a|^2+|b|^2
*how to show it?just multiply it?if yes where does||come from?
does||show that it should be a value?
| | around a number means "magnitude of". You will need to use the following relationships:
a*a = |a|^2
b*b = |b|^2
a*b = 0, because they have said that a and b are perpendicular.
Th "*" operator means "dot product", which I assume you have learned the meaning of. In general, for vectors x and y,
x*y = |x| |y| cos(theta)
where theta is the angle between the two vectors. Thus, in this case,
a) (a+2b)*(a-b)=
a*a -a*b + 2b*a -2b*b
|a|^2 - 2|b|^2
Do problem (b) the same way.
My first line should have read:
| | around a number VECTOR means magnitude of that vector
To show these equalities, we need to use the properties of dot product and the fact that a and b are perpendicular vectors.
a) Let's start with the first equality: (a + 2b) . (a - b) = |a|^2 - 2|b|^2
To solve this, we can expand the dot product using the distributive property:
(a + 2b) . (a - b) = a . (a - b) + 2b . (a - b)
Now, let's calculate each part of the expanded dot product:
a . (a - b) = |a|^2 - a . b (using the dot product property)
2b . (a - b) = -2|b|^2 + 2b . a (using the dot product property)
Now, substitute the values back into the original equation:
(a + 2b) . (a - b) = |a|^2 - a . b + (-2|b|^2 + 2b . a)
Next, rearrange the terms:
= |a|^2 + 2b . a - a . b - 2|b|^2
Now, notice that a and b are perpendicular vectors, so a . b = 0. Therefore, the equation can be simplified to:
= |a|^2 + 2b . a - 2|b|^2
Lastly, recall that a . b = 0 and substitute it into the equation:
= |a|^2 - 2|b|^2
Thus, we have shown the first equality: (a + 2b) . (a - b) = |a|^2 - 2|b|^2.
b) Now, let's move on to the second equality: (a + b) . (a - b) = |a|^2 + |b|^2
We can follow a similar approach as in part a) to solve this equation.
Expand the dot product using the distributive property:
(a + b) . (a - b) = a . (a - b) + b . (a - b)
Now, calculate each part of the expanded dot product:
a . (a - b) = |a|^2 - a . b (using the dot product property)
b . (a - b) = 0 - |b|^2 (as a and b are perpendicular, b . (a - b) = 0)
Now, substitute the values back into the original equation:
(a + b) . (a - b) = |a|^2 - a . b + (0 - |b|^2)
Simplify the equation:
= |a|^2 - a . b - |b|^2
Again, recall that a . b = 0 and substitute it into the equation:
= |a|^2 - |b|^2
Thus, we have shown the second equality: (a + b) . (a - b) = |a|^2 - |b|^2.
The double vertical bars (||) represent the magnitude or absolute value of a vector. The use of || is to signify that we are dealing with the length or magnitude of the vector, which is a numerical value. In the context of these equalities, |a| represents the magnitude of vector a and |b| represents the magnitude of vector b. So when we write |a|^2, we mean the square of the magnitude of vector a.