Instructions: On occasion, the notation = [A, θ] will be a shorthand notation for .
-If vector is added to vector , the result is a third vector that is perpendicular to and has a magnitude equal to 3 . What is the ratio of the magnitude of to that of ?
A) 1.8
B) 2.2
C) 3.2
D) 1.3
E) 1.6

Question

Instructions: On occasion, the notation  = [A, θ] will be a shorthand notation for  .
-If vector  is added to vector  , the result is a third vector that is perpendicular to  and has a magnitude equal to 3  . What is the ratio of the magnitude of  to that of  ?
A) 1.8 
B) 2.2 
C) 3.2 
D) 1.3 
E) 1.6

Quizplus · Accepted Answer

Let us first simplify the given notation as per the instructions -

11eb6adc_eda7_f9a7_b0cf_5550bdde48bd_TB8245_00 = [A, θ]  => 11eb6adc_eda8_20b8_b0cf_679ff05d6bb4_TB8245_11 = [A, θ]

Now, let's represent the given vectors as position vectors -

Vector 1 = A
Vector 2 = B = A + 11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11
Vector 3 = C = 3 * 11eb6adc_eda8_47cc_b0cf_3d7582e7a84d_TB8245_11

Since vector 2 is perpendicular to vector 11eb6adc_eda8_47cb_b0cf_93781240d26c_TB8245_11, we can use the dot product to get:

(B - A) . 11eb6adc_eda8_47cb_b0cf_93781240d26c_TB8245_11 = 0
(A + 11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11 - A) . 11eb6adc_eda8_47cb_b0cf_93781240d26c_TB8245_11 = 0
11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11 . 11eb6adc_eda8_47cb_b0cf_93781240d26c_TB8245_11 = 0

Simplifying this, we get:

cos(π/2 - θ) = 11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11 . 11eb6adc_eda8_47cc_b0cf_3d7582e7a84d_TB8245_11 / (3 * |11eb6adc_eda8_47cc_b0cf_3d7582e7a84d_TB8245_11|^2)

Since we know the magnitude of vector 3, we can solve for the magnitude of vector 11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11 in terms of θ.

|11eb6adc_eda8_20b9_b0cf_959a103ae376_TB8245_11|^2 = 3^2 * |11eb6adc_eda8_47cc_b0cf_3d7582e7a84d_TB8245_11|^2 / (cos(π/2 - θ))^2

We can now calculate the ratio of the magnitudes of vectors 11eb6adc_eda8_47cd_b0cf_3f4e14e138ca_TB8245_11 and 11eb6adc_eda8_47ce_b0cf_633e82262938_TB8245_11 using the formula:

|11eb6adc_eda8_47cd_b0cf_3f4e14e138ca_TB8245_11| / |11eb6adc_eda8_47ce_b0cf_633e82262938_TB8245_11| = sin(67.5°) / sin(112.5°) = √3 / (√3 + 1)

Simplifying this, we get:

|11eb6adc_eda8_47cd_b0cf_3f4e14e138ca_TB8245_11| / |11eb6adc_eda8_47ce_b0cf_633e82262938_TB8245_11| = (3 + √3) / (2 + √3) ≈ 1.6

Therefore, the answer is C.

Instructions: on Occasion, the Notation = [A, θ] Will