find the triple scalar product u . (v * w) for the vectors
u = 9i - 7j - 3k
v = -6i + j - 5k
w = 2i + 8j - 9k
do you mean
u dot (v X w) ????
if so
vXw =
+i +j +k
-6 +1 -5
+2 +8 -9
= (-9 +40)i + (-10-54)j +(-48-2)k
-49i -64j -50 k now dot with
+ 9i - 7j - 3 k
---> -441 + 448 + 150
157
CHECK MY ARITHMETIC
as long as we're using determinants,
u•v×w =
|9 -7 -3|
|-6 1 -5|
|2 8 -9|
To find the triple scalar product u . (v * w), we need to take the cross product of vectors v and w first and then calculate the dot product with vector u.
First, let's calculate the cross product of vectors v and w.
To calculate the cross product, we use the following formula:
v x w = (v_y * w_z - v_z * w_y)i - (v_x * w_z - v_z * w_x)j + (v_x * w_y - v_y * w_x)k
Let's plug in the given values:
v x w = ((1 * -9) - (-5 * 8))i - ((-6 * -9) - (-5 * 2))j + ((-6 * 8) - (1 * -5))k
= (-9 + 40)i - (54 - 10)j + (-48 + 5)k
= 31i - 44j - 43k
Now we have the result of the cross product: 31i - 44j - 43k.
Next, let's calculate the dot product between the resulting vector and vector u.
The formula for calculating the dot product is:
u · (v x w) = u_x * (v_x * w_x) + u_y * (v_y * w_y) + u_z * (v_z * w_z)
Let's plug in the given values:
u · (v x w) = (9 * 31) + (-7 * -44) + (-3 * -43)
= 279 + 308 + 129
= 716
Therefore, the triple scalar product u . (v * w) between the given vectors is 716.