Given u=3i-2j+k,v=2i-4j-3k, w=-i+2j+2k,
1 Find a unit vector normal to the plane containing v and w.
2 Find the volume of the parallelepiped formed by u, v, and w.
3 Are any of these vectors parallel? Orthogonal? Why or why not?
v-u x w-u is a vector perpendicular to the plane containing u,v,w. Divide by its magnitude to get a unit vector
uxv•w is the volume desired
check u•v u•w v•w for orthogonal
To solve these questions, we need to perform vector operations such as cross product and dot product.
1. To find a unit vector normal to the plane containing vectors v and w, we can take the cross product of v and w. The cross product of two vectors will result in a vector that is perpendicular (normal) to both of the original vectors.
Cross product: v x w
To find the cross product, we can use the formula:
v x w = (v_y * w_z - v_z * w_y)i + (v_z * w_x - v_x * w_z)j + (v_x * w_y - v_y * w_x)k
Substituting the given values:
v x w = ((-4 * 2) - (-3 * 2))i + ((-3 * -1) - (2 * 2))j + ((2 * 2) - (-4 * -1))k
= (-8 + 6)i + (3 - 4)j + (4 + 4)k
= -2i - j + 8k
Now, we need to find the unit vector, which is a vector with magnitude 1. To find it, we divide the cross product vector by its magnitude.
Magnitude: |v x w| = √((-2)^2 + (-1)^2 + 8^2) = √(4 + 1 + 64) = √69
Unit vector: n = (-2i - j + 8k) / √69
So, a unit vector normal to the plane containing v and w is (-2i - j + 8k) / √69.
2. To find the volume of the parallelepiped formed by vectors u, v, and w, we can use the scalar triple product.
Scalar triple product: u · (v x w)
To find the scalar triple product, we can use the formula:
u · (v x w) = u_x * (v_y * w_z - v_z * w_y) + u_y * (v_z * w_x - v_x * w_z) + u_z * (v_x * w_y - v_y * w_x)
Substituting the given values:
u · (v x w) = (3 * (-4 * 2 - (-3 * 2))) + (-2 * (-3 * -1 - (2 * 2))) + (1 * (2 * 2 - (-4 * -1)))
= (3 * (-8 + 6)) + (-2 * (3 - 4)) + (1 * (4 + 4))
= (-6) + (2) + (8)
= 4
So, the volume of the parallelepiped formed by vectors u, v, and w is 4.
3. To determine if any of these vectors are parallel or orthogonal, we can use the dot product.
If the dot product of two vectors is zero, they are orthogonal (perpendicular). If the dot product is equal to the product of their magnitudes, then they are parallel.
Dot product: u · v
To find the dot product, we can use the formula:
u · v = u_x * v_x + u_y * v_y + u_z * v_z
Substituting the given values:
u · v = (3 * 2) + (-2 * (-4)) + (1 * (-3))
= 6 + 8 - 3
= 11
Since the dot product of u and v is not zero and is not equal to the product of their magnitudes, u and v are neither orthogonal nor parallel.
Similarly, you can calculate the dot product for other pairs of vectors (u, w) and (v, w) to determine if they are parallel or orthogonal using the same process.
Note: In this case, we have already calculated the dot product of u and v.