Find any vector w that is perpendicular to both vector "u = 3j + 4k" and vector "v = 2i".
Note: i, j and k are unit vectors
How would you solve this problem? Please walk me through?
Two vectors are perpendicular if their dot product is zero
so you want a vector w = (a,b,c) so that
(a,b,c)∙(0,3,4) = 0 and
(a,b,c)∙(2,0,0) = 0
from the second 2a = 0, so a=0
from the first 3b+4c=0
or 3b=-4c
b/c = -4/3, if we let c=3 and b=-4 that would work
Notice there is more than one solution
as long as our vector w has direction (0,-4,3)
Well, let's see, we need to find a vector that is perpendicular to both vector u = 3j + 4k and vector v = 2i.
To do that, we can use the cross product. The cross product of two vectors gives us a vector that is perpendicular to both of them.
First, let's find the cross product of u and v. We can write u as (0)i + (3)j + (4)k and v as (2)i + (0)j + (0)k.
Now, we can calculate the cross product by using the formula:
(u x v) = (aybz - azby)i - (axbz - azbx)j + (axby - aybx)k
Let's plug in the values:
(u x v) = (3*0 - 4*0)i - (0*0 - 4*2)j + (0*2 - 3*0)k
Simplifying, we get:
(u x v) = 0i + 8j + 0k
So, the cross product of u and v is 0i + 8j + 0k.
This means that any vector that is a multiple of this result will be perpendicular to both u and v.
Therefore, a vector w that is perpendicular to both u and v can be written as w = 0i + Aw + 0k, where A is any non-zero scalar.
In other words, w can be any vector of the form w = (0)i + (Aw)j + (0)k, where A is any non-zero scalar.
Hope that helps! Let me know if you need any more assistance or if you'd like a joke to lighten the mood!
To find a vector perpendicular to both vector u and vector v, we can use the cross product.
1. First, let's define vector u and vector v:
- u = 3j + 4k
- v = 2i
2. To take the cross product of two vectors, we will use the formula:
w = u x v
3. Now, let's compute the cross product:
- The cross product of u and v can be calculated by the determinant:
| i j k |
| 0 3 4 |
| 2 0 0 |
- Expanding the determinant, we get:
w = (3 * 0 - 4 * 0)i - (0 * 0 - 4 * 2)j + (0 * 0 - 3 * 2)k
= 0i - (-8)j - 6k
= 8j + 6k
4. Therefore, the vector w = 8j + 6k is perpendicular to both vector u = 3j + 4k and vector v = 2i.
Note: The cross product of two vectors always yields a vector perpendicular to both of them.
To find a vector perpendicular to two given vectors, we can use the cross product of the two vectors.
1. Begin by computing the cross product of the given vectors u and v.
The cross product of two vectors can be found using the determinant of a matrix. We can write the expression as follows:
w = u x v = | i j k |
| 3 4 0 |
| 2 0 0 |
By expanding the determinant, we get:
w = (4 * 0 - 0 * 0) i - (3 * 0 - 2 * 0) j + (3 * 0 - 2 * 4) k
2. Simplify the expression:
w = 0i - 0j - 8k
w = -8k
Thus, w = -8k is a vector perpendicular to both u = 3j + 4k and v = 2i.