Given three vectors

a⃗ =−i⃗ −4j⃗ +2k⃗
,
b⃗ =3i⃗ +2j⃗ −2k⃗
,
c⃗ =2i⃗ −3j⃗ +k⃗
, calculate
a⃗ ⋅(b⃗ ×c⃗ )

Given

a=<-1,-4,2>
b=<3,2,-2>
c=<2,-3,1>

bxc=
i j k
3 2-2
2-3 1
=<-4,-7,-13>
so
a.(bxc)
=<-1,-4,2>.<-4,-7,-13>
=4+28-26
=6

To calculate the dot product of the vector a⃗ with the cross product of vectors b⃗ and c⃗, you can follow these steps:

Step 1: Calculate the cross product of vectors b⃗ and c⃗.
The cross product of two vectors is given by the formula:
b⃗ × c⃗ = (b₂c₃ - b₃c₂)i + (b₃c₁ - b₁c₃)j + (b₁c₂ - b₂c₁)k

We can substitute the values of vectors b⃗ and c⃗ into the formula and evaluate:
b⃗ × c⃗ = ((3)(-3) - (2)(1))i + ((-2)(2) - (3)(2))j + ((3)(-3) - (3)(2))k
= (-9 - 2)i + (-4 - 6)j + (-9 - 6)k
= -11i - 10j - 15k

Step 2: Calculate the dot product of vector a⃗ and the cross product of b⃗ and c⃗.
The dot product of two vectors is given by the formula:
a⃗ ⋅ b⃗ = a₁b₁ + a₂b₂ + a₃b₃

We can substitute the values of vector a⃗ and the calculated cross product into the formula and evaluate:
a⃗ ⋅ (b⃗ × c⃗ ) = (-1)(-11) + (-4)(-10) + (2)(-15)
= 11 + 40 - 30
= 21

Therefore, a⃗ ⋅ (b⃗ × c⃗ ) = 21.