Let x, y, and z be vectors in R³ such that
(i) z = 3x
(ii) y - x = 4
(iii) z × y = 4e₁ (standard basis vector); and
(iv) ||x|| = 5.
We are asked to determine the value of z· (2y+z) + || (x + 2y) × z||.
First, let's simplify the expression z· (2y+z):
z· (2y+z) = z·2y + z·z
Using the distributive property of the dot product, we can rewrite this as:
z· (2y+z) = 2(z·y) + z·z
Next, let's find the value of z·y:
From equation (ii), we know that y - x = 4. Rearranging this equation, we get:
y = x + 4
Substituting this into equation (i), we have:
z = 3x
Substituting z = 3x into y = x + 4, we get:
z·y = 3x·(x + 4)
Expanding this using the distributive property, we have:
z·y = 3(x·x + 4x)
Using the property that x·x = ||x||², we can rewrite this as:
z·y = 3(||x||² + 4x)
Since we know that ||x|| = 5, we can substitute this into the equation:
z·y = 3(5² + 4x)
Simplifying further, we have:
z·y = 3(25 + 4x)
Expanding this, we get:
z·y = 75 + 12x
Now, let's find the value of z·z:
From equation (i), we know that z = 3x. Substituting this into z·z, we have:
z·z = (3x)·(3x)
Using the property that (a·b)² = ||a||² ||b||² cos²θ, where θ is the angle between a and b, we can rewrite this as:
z·z = (3² ||x||²) cos²θ
Since ||x|| = 5, we can substitute this into the equation:
z·z = (3² 5²) cos²θ
Simplifying further, we have:
z·z = 9(25) cos²θ
z·z = 225 cos²θ
Now, let's find the value of || (x + 2y) × z||:
Using the distributive property of the cross product, we can rewrite this as:
|| (x + 2y) × z|| = ||x × z + 2y × z||
Since we know that z = 3x, we can substitute this into the equation:
|| (x + 2y) × z|| = ||x × 3x + 2y × 3x||
Expanding this using the distributive property, we have:
|| (x + 2y) × z|| = ||3x × x + 3x × 2y||
Using the property that a × b = -b × a, we can rewrite this as:
|| (x + 2y) × z|| = ||-x × 3x + 2y × 3x||
Expanding this further, we get:
|| (x + 2y) × z|| = ||-3x × x + 6y × x||
Using the property that a × a = 0, we can simplify this to:
|| (x + 2y) × z|| = ||6y × x||
Using the property that ||a × b|| = ||a|| ||b|| sinθ, where θ is the angle between a and b, we can rewrite this as:
|| (x + 2y) × z|| = 6 ||y|| ||x|| sinθ
Since ||x|| = 5, we can substitute this into the equation:
|| (x + 2y) × z|| = 6 ||y|| (5) sinθ
Simplifying further, we have:
|| (x + 2y) × z|| = 30 ||y|| sinθ
Now, let's put it all together:
z· (2y+z) + || (x + 2y) × z|| = 2(z·y) + z·z + || (x + 2y) × z||
Substituting the values we found earlier:
z· (2y+z) + || (x + 2y) × z|| = 2(75 + 12x) + 225 cos²θ + 30 ||y|| sinθ
This is the final expression.