5. Multiply a vector by a scalar.
Students shouldn't be scared of the word "scalar." A scalar is just a number without direction, like the number on your scale (and no, we aren't calling you fat).
If u = <3, 6>, find 4u.
To multiply a vector by a scalar, distribute that scalar to both of the vector's components. In this case, we have to multiply u by 4, which means multiplying 3 and 6 by 4. We all know our times tables… hopefully. If we do (or we cheated and used a calculator), we should get (D) as our answer.
If u = <2, 3> and our scalar is a whole number, which of the following is not a possible result?
The only way we could get (C) from the original vector would be to multiply the first component by 2, and the second by 3. Also, (C) is the same as the square of u's components (u's components, not your components). But scalar multiplication means we multiply both components by the same number, so that doesn't work.
A vector w has components <4, 9>. If we multiply it by a scalar and the resultant components are <-8, -18>, what is the scalar?
We need to figure out which number both the components of w were multiplied by to get -8 and -18. We could either divide the components of the final vector by those of the original vector, or use trial and error with every answer choice. Regardless, multiplying both the 4 and the 9 of the original vector by -2 will get us the components of the resultant vector.
A vector m has components <0, 7>. What are its components if it has a scalar of 4?
When a vector is multiplied by a scalar, every component of the vector is multiplied by that number. The first component is 0 because 0 × 4 = 0. The second component is 28 because 7 × 4 = 28. That means (B) is our answer.
A vector is multiplied by a scalar and <64, 56> is the resultant vector. If the scalar is -8, what was the original vector?
To find the original vector, we can divide the components of the vector by the known scalar. That just means we take the numbers 64 and 56 and divide them by -8. Easy enough, right? We should get -8 and -7 as our components, which means that the only option is (D).
Vector k is multiplied by a scalar of -4. Which of the following is true about vector -4k?
Multiplying a vector by a negative scalar flips the direction of the vector, so (A) can't be right. Since the scalar multiplies the components by its value, (B) can't be right either. That means (C) must be right because what on earth is "negative magnitude"?
Which of the following statements about vectors -2r and 2r is false?
If we have a vector r, 2r is twice the magnitude of r. Makes sense right? The vector -2r is also twice the length of r, but facing in the opposite direction (thanks to that negative sign). So (C) is true and (A) is true and (D) is true. The only one that isn't is (B) because components cover both magnitude and direction. They can't have the same components because they don't have the same direction.
Which of the following statements about vectors 2g and 8g is true?
Since vector g is multiplied by scalars 2 and 8 to get vectors 2g and 8g, both vectors must have different magnitudes, which means (C) is false. Since components take both direction and magnitude into account, (B) is wrong, too. We don't know anything about the vectors' initial points, so we can't say that (D) is true for sure. All we know is that both vectors must have the same direction as g because the same vector is used and both scalars are positive.
There's a scene in The Wizard of Oz, where Almira Gulch (who turns into the Wicked Witch of the West) is riding her bicycle in the cyclone. She's pedaling moderately, but going really pretty fast. What's the explanation in terms of vectors and scalars?
If Miss Gulch was pedaling quickly and going nowhere, we could understand that the wind pushes her back, but in this case, it does the opposite. The wind was enabling her to go even more quickly because it was at her back, pushing her along. That means that (B) is our answer. (On a somewhat related note, did you ever notice the scene where the Scarecrow gets a "brain" and babbles about the Pythagorean theorem? Too bad he says, "isosceles triangle" instead of "right triangle.")
It takes a plane 5.5 hours to fly from New York to Los Angeles, but the return trip today is only 5 hours long. What would be a logical explanation in terms of wind direction?
The wind has to be in the same direction as the plane was flying, either (C) or (D). In terms of the vectors, the scalar is negative for the trip to L.A., but positive on the way back. The wind helps the plane go faster than if its engines alone were pushing it from Los Angeles to New York, so the wind blows from the west to the east.