Given a=(2,1,0), b=(-1,0,3), c=(4,-1,1), calculate the following triple scalar and triple vector products.
a x b•c
AAAaannndd the bot gets it wrong yet again!
a×b = (3,-6,1)
a×b•c = 19
I apologize for the mistake in my previous response. Thank you for correcting me.
You are correct that a x b = (3, -6, 1).
To find a x b • c, we first need to find the cross product of a x b, which we have already calculated as (3, -6, 1).
Next, we take the dot product of the resulting vector with c:
(3, -6, 1) • (4, -1, 1) = (3)(4) + (-6)(-1) + (1)(1) = 19
Therefore, a x b • c = 19.
To calculate the triple scalar product (a x b•c), we need to first calculate the cross product of vectors a and b, and then take the dot product of the resulting vector with vector c.
Let's start by calculating the cross product of vectors a and b:
a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
Substituting the values of vectors a and b in the formula, we have:
a x b = ((1*3) - (0*(-1)), (0*(-1)) - (2*3), (2*0) - (1*(-1)))
Simplifying:
a x b = (3 + 0, 0 - 6, 0 - (-1))
a x b = (3, -6, 1)
Now, let's take the dot product of the resulting vector with vector c:
(a x b) • c = (3*4) + (-6*(-1)) + (1*1)
Simplifying:
(a x b) • c = 12 + 6 + 1
(a x b) • c = 19
Therefore, the triple scalar product (a x b•c) is equal to 19.
The triple vector product is given by:
a x b x c = (a . c) b - (a . b) c
where "." denotes the dot product.
First, we find the vector products:
a x b = (2, 1, 0) x (-1, 0, 3) = (-3, -6, -1)
b x c = (-1, 0, 3) x (4, -1, 1) = (-3, 11, 1)
Next, we find the dot products:
a . b = (2)(-1) + (1)(0) + (0)(3) = -2
a . c = (2)(4) + (1)(-1) + (0)(1) = 7
Putting it all together, we get:
a x b x c = (7)(-1, 0, 3) - (-2)(4, -1, 1)
= (-7, 14, -17)
Therefore, the triple vector product is (-7, 14, -17).
To find the triple scalar product, we take the dot product of a x b with c:
(a x b) . c = (-3, -6, -1) . (4, -1, 1)
= (-3)(4) + (-6)(-1) + (-1)(1)
= -12 + 6 - 1
= -7
Therefore, the triple scalar product is -7.