On occasion, the notation will be a shorthand notation for .
-When vector is added to vector , which has a magnitude of 5.0, the vector representing their sum is perpendicular to and has a magnitude that is twice that of . What is the magnitude of ?
A)2.2
B)2.5
C)4.5
D)5.0
E)7.0

Question

On occasion, the notation  will be a shorthand notation for  .
-When vector  is added to vector  , which has a magnitude of 5.0, the vector representing their sum is perpendicular to  and has a magnitude that is twice that of  . What is the magnitude of  ?
A)2.2 
B)2.5 
C)4.5 
D)5.0 
E)7.0

Quizplus · Accepted Answer

First, we can use the given shorthand notation to simplify the problem. We have:

- Vector A: 11eb8c7e_f55d_43aa_8371_759596490baa_TB8847_11 
- Vector B: 11eb8c7e_f55d_43ab_8371_5f8d58b08edb_TB8847_11 
- Vector C: 11eb8c7e_f55d_43ac_8371_a344195b461c_TB8847_11 
- Vector D: 11eb8c7e_f55d_6abd_8371_7bc36b947231_TB8847_11 
- Vector E: 11eb8c7e_f55d_6abe_8371_9f424b2e7a98_TB8847_11

Next, we need to use the given information to set up equations. Let x be the magnitude of vector A.

- Since vector A and vector B have the same first three components, their magnitudes are equal. Therefore, the magnitude of vector B is also x.
- We know that the sum of vector A and vector B is perpendicular to vector C. This means that their dot product is zero:

(A + B) · C = 0

Expanding the dot product using the given coordinates, we get:

(x + x) * (a + b + 8e + c + 7d + 2f + 8b + 8) = 0

Simplifying this expression, we get:

4x + 14 = 0

Solving for x, we get:

x = -3.5

This doesn't make physical sense, since magnitudes are always positive. Therefore, we made an error somewhere in our calculations. We can try using the other given information to check our work:

- We know that the sum of vector A and vector B has a magnitude that is twice that of vector D. This can be expressed as:

||A + B|| = 2 ||D||

Substituting the magnitudes of A, B, and D, we get:

2(x + x) = 2(5)

Simplifying this expression, we get:

x = 2.5

This is a valid magnitude, so we can use it to find the magnitude of vector E:

||E|| = ||A + B + D|| = ||A + B|| + ||D||

Substituting the magnitudes of A, B, and D, we get:

||E|| = 2(x) + 5

Plugging in x = 2.5, we get:

||E|| = 5 + 5 = 10

Therefore, the magnitude of vector E is 10, which corresponds to choice (A).

On Occasion, the Notation Will Be a Shorthand Notation