Test: Vectors and Scalars - MCAT MCQ

Let’s consider each case graphically to narrow down our choices. Recall that vectors  are half the magnitude of vector  By drawing out  we can see that the displacement is somewhat large:

All three vectors have the same magnitude here. By drawing out , we can see that displacement is not large, but could be closer to zero:

Between the remaining two, it becomes hard to discern visually which has the least displacement:


In analyzing  let’s pick 10 for the magnitude of each vector, and resolve all the vectors into the x and y direction. In a 45 − 45 − 90 triangle, remember that the sides are in a proportion of 1 : 1 : √2 so with the hypotenuse as 10 units, the x and y components are 5√2, units. In the x-direction, the x component vectors  overlap and add up to 10 √2, which is opposed by vector , with, vector, on top, with a magnitude of 10.

In the y-direction, the y-component of vectors cancel each other out. The resultant vector points in the negative y-direction with an approximate magnitude of 4 since √2, is approximately equal to 1.4.

In analyzing  let’s pick 10 for the magnitude of vectors  and 5 for the magnitude of vector  resolve all the vectors into the x and y components. Vector  resolves into an x and y components with magnitudes of 5√2 units:

The resultant x-component is 2 in the negative-x direction, and the resultant y-component is 3 in the negative-y direction. To find our displacement, use the Pythagorean theorem to obtain √13 which is less than 4.

This question can be solved visually and without any math. Let’s go through drawing each vector combination.
The vector combination  can be drawn like the following:

The vector combination  can be drawn like the following:

The vector combination  can be drawn like the following:

The vector combination  can be drawn like the following:

Visually, we can eliminate down to  and since is much longer than  the vector combination  will have the greatest displacement.

A scalar multiplied by a scalar will produce another scalar. For instance, distance divided by time is equal to speed, which is a scalar. In the case of power, energy divided by time is equal to power. Statement III is true.
A vector multiplied by a scalar is representative of the scalar product or the dot product and will always produce a vector. The scalar product of two vectors  can be constructed by taking the component of  in the direction of  and multiplying it times the magnitude of  and it can be expressed as AB cos θ.

Torqueis not an example of the scalar product of vectors since it is equal to rF sinθ. A correct example of the dot product would be work, which is equal to Fd cos θ. Therefore, statement I is not true.
Vector multiplied by a vector is representative of the vector product or the cross product and will produce a vector or scalar. The magnitude of the vector product of  can be constructed by taking the product of the magnitudes of multiplied by the sine of the angle between them.
Magnetic force is an example of the vector cross product, so statement II is true. Therefore, the correct answer is that statements II and III are true.

In identifying a scalar quantity, look for a quantity that only has magnitude, but a more surefire method is to find a quantity that consists of a scalar multiplied by a scalar. These, of course, would be the wrong answers.

Power is energy divided by time, which is a scalar divided by a scalar. Hence, it is a scalar:
P = E/t

Density is mass divided by volume, which also is a scalar divided by a scalar. Hence, it is a scalar:
ρ = m/V

Capacitance is charge divided by voltage, which also is a scalar divided by a scalar. Hence, it is a scalar:

C = Q/V

Torque does have a direction, either counterclockwise or clockwise, and is level arm multiplied by force, which is a scalar multiplied by a vector. Hence, it is a vector, and torque is the correct answer.
τ = rF sinθ

In identifying a vector quantity, look for a quantity that has both direction and magnitude, but a more surefire method is to find a quantity that consists of a vector multiplied by a scalar. The vector quantities are the wrong answers.

Force is mass multiplied by acceleration, which is a scalar multiplied by a vector. Hence, it is a vector:
F = ma

Momentum is mass multiplied by velocity, which is a scalar multiplied by a vector. Hence, it is a vector:
p = mv

Electric field is force divided by charge, which is a vector divided by a scalar. Hence, it is a vector:
E = F/q

Electric potential is electric potential energy divided by charge, which is a scalar divided by a scalar. Hence, it is a scalar, and electric potential is the correct answer.
V = U/q

For two vectors  the vector sum   is obtained by placing them head to tail and drawing the vector from the free tail to the free head. A vector difference is equivalent to a vector sum with the orientation of the second vector reversed.
This vector combination below best represents the resultant vector 

This vector combination was probably intended to represent but it doesn't follow the head-to-tail rule.

This vector combination was probably intended to represent  but it doesn't follow the head-to-tail rule.

This vector combination below can be represented by 

In each of these combinations, vector  must be flipped for the subtractive process. Each one must be drawn out to decide what one has the greatest magnitude.
Once vector A is inverted, the tail of vector  should be placed at the head of vector  to obtain the resultant vector. The vector combination only shows the greatest displacement in the x-direction, which is 15 units:

This vector combination shows the greatest displacement in the y-axis, but in either axes, there is loss of displacement, so in either direction it would not surpass 15 units from the previous hint:

This vector combination shows comparable displacement in the x- and y-axes. In the horizontal, it will be some value less than 15 units, and in the vertical, it will be some value greater than 10 units:

This vector combination shows the greatest displacement in y-axis. The y-component of each will definitely add to some value greater than 15 units:

The last vector combination will show the greatest displacement in the y-axis.

In order to cross the river at a point directly across the street, the ferry must launch upstream:

While this may not be immediately apparent, understand that 1.4, point, 4 is equal to √2. This creates a relationship for a 45 - 45 - 90 triangle, where the ratio of the sides is 1 : 1 : √2. Therefore, the ferry will launch at an angle of 45°, degrees from the shore.

As a result, the velocity of the boat relative to the earth is 1.0 m/s. Traveling 1600 meters at that speed, not 10 meters per second, will take 1600 seconds or 27 minutes.

The correct answer is that the ferry must launch upstream at an angle of 45° from the shore with a velocity of 1.4 m/s.

Given this formula, we have to extrapolate that charge q is directly proportional to F,  while r has an inverse-square relationship to F.

Each charge will exert a force on charge A, and their force vectors are in the diagram below:

If the force of charges B and D on charge A are each 1 unit, then the force of charge C will be 1/2 unit. The doubling of the charge on C is balanced by the doubling of the distance, but distance is squared, so the ratio becomes

The resultant vector would be the following:

Aisha is pulling with 20 N, so in order for this system to be in equilibrium, Saul and Lorenzo together must be pulling with the same force in the opposite direction. That vector would represent the hypotenuse of the right triangle where Saul’s force is the adjacent side and Lorenzo’s opposite.

One big clue to solving this problem is that the angle is equal to arctan (1.33). 1.33 is the value of the opposite side divided by the adjacent side.

Let’s represent the decimal by a fraction, which would be 4/3. So the ratio of the opposite side to the adjacent side is 4 : 3.

Either by Pythagorean theorem or by past memorization, the hypotenuse will have the value of 5, and we have our 3 - 4 - 5 triangle.

As we said, Aisha represents the hypotenuse, which has a value of 20 N. Divide that by the calculated value for the hypotenuse to obtain the factor to multiply the other sides. That factor is 4.

The other sides of the triangle are 12 N and 16 N. Match it to the right person. The correct answer is 16 N for  Lorenzo and 12 N for Saul.