Dot product and cross product evaluation questions

In summary, we discussed how to evaluate the expressions (r.t)s and (s.t)r, and deduced that they are equal to each other. We also discussed the dot/scalar product and how to evaluate it using given vectors. Finally, we evaluated the expression (r.t)s - (s.t)r and found that it is equal to (r x s) x t. We then discussed how this holds true for any three vectors.
  • #1
thomas49th
655
0
This question has a few parts.

r = i + 2j + 3k
s = 2i - 2j - 5k
t = i - 3j - k

Evaluate:

a)(r.t)s - (s.t)r
b)(r x s) x t.

deduce that (r.t)s - (s.t)r = (r x s) x t


can you prove this relative true for any three vectors

a)(r.t)s - (s.t)r
(r.t)s

well I don't know what s is doing to inside the bracket. I don't think it's the cross product rule.
Anyhow

(r.t) means use dotty dot product

r.t = |r||t|cos(x)

problem uno. I don't know the angle between the vectors.

Am I just being stupid. I thought about possibly trying to get the angle from the cross product, or using some trig identity but I think that'll be a road to nowhere

Hint?

Thanks
Thomas
 
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  • #2
hi,

ok, first you need: (a) (r.t)s - (s.t)r

you are given the vectors. For this, you don't need to use the formula:
r.t = |r||t|cos(x)

If you have two vector, do you know how to evaluate the dot/scalar product?
 
  • #3
im being an idiot

r.t = (i - 6j- 3k)

but what operation does the s outside of (r.t)s do?

Thanks
 
  • #4
and I am being an idiot agin, the dot product is scalar and so I'm merely multiplying all values of s by the scalar value returns from r.t

1 - 6 - 3 = -8

so (r.t)s = -8(s)

=> s = 16i - 16j - 40k

Good so far?
 
  • #5
ya, good so far :)

now, do the same for (s.t)r

so can you now evaluate (a) (r.t)s - (s.t)r
 

1. What is the dot product?

The dot product, also known as the scalar product, is a mathematical operation used to calculate the product of two vectors. It results in a scalar value and is calculated by multiplying the corresponding components of the vectors and adding them together.

2. How is the dot product useful?

The dot product is useful in various fields of science, technology, and engineering. It can be used to calculate the angle between two vectors, determine the projection of one vector onto another, and solve equations involving vectors.

3. What is the cross product?

The cross product, also known as the vector product, is a mathematical operation used to calculate the product of two vectors. It results in a vector that is perpendicular to both input vectors and its magnitude is equal to the area of the parallelogram formed by the two vectors.

4. How is the cross product used in real-world applications?

The cross product has various applications in physics, engineering, and computer graphics. It is used to calculate torque, magnetic force, and angular momentum in physics. In engineering, it is used to determine the direction of a force in a three-dimensional space. In computer graphics, it is used to simulate lighting and shading effects.

5. What is the difference between the dot product and the cross product?

The main difference between the dot product and the cross product is the type of result they produce. The dot product results in a scalar value, while the cross product results in a vector. Additionally, the dot product is commutative, meaning the order of the input vectors does not matter, while the cross product is not commutative, meaning the order of the input vectors affects the direction of the resulting vector.

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