The table gives the probabilities that new cars of different models will have brake failure. The car model that is least likely to have a brake failure is model , and the probability of brake failure for this model is %.Car Model            Brake Failure
                             in New Car
A                            0.0065%
B                            0.0037%
C                            0.0108%
D                            0.0029%
E                            0.0145%
Total 0.0048%

(a) A + B: The result is -4i - 6j, with a magnitude of 2√13 and a direction angle of arctan(3/2).

(b) A - B: The result is -2i + 6j, with a magnitude of 2√10 and a direction angle of arctan(-3).

The given vectors are:

B = -i - 4j

A = -3i - 2j

(a) A + B:

To add two vectors,

Simply add their corresponding components.

So, A + B = (-3i - 2j) + (-i - 4j)

Combining the i-components, we get:

-3i - i = -4i.

And combining the j-components, we get:

-2j - 4j = -6j.

Therefore, A + B = -4i - 6j.

To find the magnitude of A + B,

Use the Pythagorean theorem:

|A + B| = ,

Where x is the magnitude of the i-component and y is the magnitude of the j-component.

In this case,

x = -4 and y = -6,

So: |A + B| =

|A + B| = 2√13

To find the direction angle of A + B,

Use the arctan function:

θ = arctan(y / x),

Where y is the j-component and x is the i-component.

In this case,

x = -4 and y = -6,

so: θ = arctan(-6 / -4)

θ = arctan(3/2)

Therefore, the magnitude of A + B is 2√13 and the direction angle is arctan(3/2).

(b) A - B:

To subtract two vectors,

Subtract their corresponding components.

So, A - B = (-3i - 2j) - (-i - 4j).

Combining the i-components, we get:

-3i + i = -2i.

And combining the j-components, we get:

-2j - (-4j) = 2j + 4j = 6j.

Therefore, A - B = -2i + 6j.

To find the magnitude of A - B,

Use the Pythagorean theorem:

,

Where x is the magnitude of the i-component and y is the magnitude of the j-component.

In this case,

x = -2 and y = 6,

so:

|A - B| = 2√10

To find the direction angle of A - B,

Use the arctan function:

θ = arctan(y / x),

Where y is the j-component and x is the i-component.

In this case, x = -2 and y = 6,

so:

θ = arctan(6 / -2)

θ = arctan(-3)

Therefore,

The magnitude of A - B is 2√10 and the direction angle is arctan(-3).

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The complete question is:

For vectors B =−i −4j  and A =−3i −2j ,

Calculate (a) A + B and its magnitude and direction angle, and (b) A − B and its magnitude and direction angle.

The resulting vectors after adding and subtracting vectors A and B are A + B = -4i - 6j with a magnitude of 7.21 and A - B = -2i + 2j with a magnitude of 2.83. The direction angle for the vectors are calculated using arctan of the absolute value of the components' ratios.

For vectors B = -i - 4j and A = -3i - 2j, you first need to add and subtract these vectors component-wise to get the resulting vectors. Addition gives A + B = -i + (-3i) + -4j + (-2j) = -4i - 6j, whereas subtraction gives A - B = -3i - (-i) + -2j - (-4j) = -2i + 2j.

The magnitude of a vector is calculated by the Pythagorean theorem: (magnitude of A+B) = sqrt((-4)^2 + (-6)^2) = 7.21 and (magnitude of A-B) = sqrt((-2)^2 + 2^2) = 2.83.

The direction angle is found using arctan(|Ay / Ax|), but adjustment must be made depending on the quadrant of the resulting vector. In conclusion, (a) A + B = -4i - 6j, |A + B| = 7.21, angle = arctan(6/4), and (b) A - B = -2i + 2j, |A - B|= 2.83, angle = arctan(2/2).

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Answer:

Answer is attached to steps.

Step-by-step explanation:

Those are the steps and the answer attached