If = +4 î - 2 ĵ - 3
And = -4 î -2 ĵ - 3 , which of the following numbers is closest to the magnitude of - ?
A) 8
B) 7
C) 9
D) 10
E) 11

Question

If  = +4 î - 2 ĵ - 3 
And  = -4 î -2 ĵ - 3  , which of the following numbers is closest to the magnitude of  -  ?
A) 8 
B) 7 
C) 9 
D) 10 
E) 11

Quizplus · Accepted Answer

Given that 11ea98f3_0f6b_338c_99aa_d7be98803997_TB2997_11 = +4 î - 2 ĵ - 3 11ea98f3_0f6b_338d_99aa_4dddf04ebadb_TB2997_11 and 11ea98f3_0f6b_338e_99aa_33d9e8708dd1_TB2997_11 = -4 î -2 ĵ - 3 11ea98f3_0f6b_5a9f_99aa_65fae1394307_TB2997_11, we can see that the vector difference between these two is 8 î.

Now, we are asked to find the closest magnitude between 11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11 - 11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11. Let's simplify this vector difference as V.

We can apply the triangle inequality to get an upper bound on the magnitude of V:

|V| ≤ |11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11| + |-11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11|

We can then use the given information that 11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11 - 11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11 = 8 î to simplify further:

|V| ≤ |11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11| + |11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11 - 11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11|

|V| ≤ |11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11| + 8

The magnitude of 11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11 can be simplified as:

|11ea98f3_0f6b_5aa1_99aa_a1734dd292e1_TB2997_11| = sqrt((-4)^2 + (-2)^2 + (-3)^2) = sqrt(29)

We don't have enough information to directly calculate |11ea98f3_0f6b_5aa0_99aa_6dbcbf28155d_TB2997_11|, but we can use the fact that it is closest to the origin of coordinate system x-y. Based on this fact, we can guess that it is approximately equal to the magnitude of 11ea98f3_0f6b_338c_99aa_d7be98803997_TB2997_11, which is sqrt(4^2 + 2^2 + 3^2) = sqrt(29).

Putting all of this together, we get:

|V| ≤ sqrt(29) + 8

The closest integer to this quantity is 8, so the answer is (A).