Solve (d1 + d2) . (d1 x 4d2): Unit Vector Question

In summary, the problem is asking for the scalar triple product of (d1 + d2) and (d1 x 4d2). To solve it, you need to add (d1 + d2), perform a cross product of (d1 x 4d2), and then take the dot product of the results. This is known as the scalar triple product.
  • #1
Shatzkinator
53
0

Homework Statement


If d1 = 3i - 2j +4k and d2 = -5i + 2j -k, then what is (d1 + d2) . (d1 x 4d2)?


Homework Equations


c = absin(theta) --> vector product
c = abcos(theta) --> scalar product

The Attempt at a Solution


I looked at a sample problem and they show the distributive law for components, however one of the calculations was 3i x 3k = 9(-j)... how does that work (ie. using the above formula does not take into account any of the letters or unit vectors or whatever)? Second, how do you know if its vector or scalar product. Thanks a bunch...
 
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  • #2
Shatzkinator said:

Homework Statement


If d1 = 3i - 2j +4k and d2 = -5i + 2j -k, then what is (d1 + d2) . (d1 x 4d2)?

Homework Equations


c = absin(theta) --> vector product
c = abcos(theta) --> scalar product

The Attempt at a Solution


I looked at a sample problem and they show the distributive law for components, however one of the calculations was 3i x 3k = 9(-j)... how does that work (ie. using the above formula does not take into account any of the letters or unit vectors or whatever)? Second, how do you know if its vector or scalar product. Thanks a bunch...

Welcome to PF.

What you basically have is the scalar triple product.

(d1 + d2) dot (d1 x 4d2)

To resolve it you need to first add the (d1 + d2) term.
Then perform the Cross Product of (d1 x 4d2).
Then the Dot product of the results of the first 2 steps.

http://en.wikipedia.org/wiki/Scalar_triple_product#Scalar_triple_product
 
  • #3


As a scientist, it is important to understand the concepts and equations being used in a problem before attempting to solve it. In this case, the problem involves both vector and scalar products, which have different equations and meanings.

First, let's define the terms used in this problem. d1 and d2 are vectors, which are quantities that have both magnitude and direction. In this case, they are given in terms of unit vectors i, j, and k, which represent the x, y, and z directions. The dot product (represented by a dot between two vectors) is a scalar product, which results in a single value and represents the magnitude of the projection of one vector onto the other. The cross product (represented by a cross between two vectors) is a vector product, which results in a vector and represents the direction and magnitude of a vector perpendicular to both of the original vectors.

Now, let's solve the problem. First, we need to find the sum of d1 and d2, which is (3i - 2j + 4k) + (-5i + 2j - k) = -2i + 0j + 3k. Next, we need to find the cross product of d1 and 4d2, which is (3i - 2j + 4k) x (-20i + 8j - 4k) = -32i - 28j + 25k. Finally, we take the dot product of these two results: (-2i + 0j + 3k) . (-32i - 28j + 25k) = -64 + 0 + 75 = 11.

To answer your questions, the distributive law for components is used to simplify the calculations by distributing the operations to each component of the vectors. In the example you mentioned, 3i x 3k = 9(-j) uses the fact that i x k = j, so the i and k components are multiplied together and the j component is multiplied by the product of the other two components. This is a simplified version of the full cross product equation.

To determine if it is a vector or scalar product, you can look at the result. If it is a single value, it is a scalar product. If it is a vector, it is a vector product. In this case, the final result is a single value, so it
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is used to represent a specific direction in space. It is typically denoted by a hat symbol (^) on top of the vector symbol.

2. How do you solve the equation (d1 + d2) . (d1 x 4d2)?

To solve this equation, you will need to first find the dot product of (d1 + d2) and (d1 x 4d2). This can be done by multiplying the corresponding components of the two vectors and then adding them together. The resulting scalar value will be the solution to the equation.

3. What are the units for the solution to this equation?

The units for the solution to this equation will depend on the units of the two vectors involved. If the units for d1 and d2 are the same, the resulting unit will be the square of that unit. If the units for d1 and d2 are different, the resulting unit will be the product of the two units.

4. How do you represent a unit vector in notation?

A unit vector is typically represented in notation by placing a hat symbol (^) on top of the vector symbol. For example, a unit vector in the x-direction would be denoted as ^i, a unit vector in the y-direction would be denoted as ^j, and a unit vector in the z-direction would be denoted as ^k.

5. What is the significance of using a unit vector?

Unit vectors are important in mathematics and physics because they allow us to represent and manipulate directional quantities without having to worry about their magnitude. This simplifies calculations and allows for a more intuitive understanding of vector operations.

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