Choose a representation
Compute each component separately. Scalar multiplication reaches every entry; a two-by-two determinant uses a difference of diagonal products. A magnitude discards direction and is nonnegative.
Worked example: follow the quantity
Vectors u=(3,−2) and w=(3,5) are given. What is the second component of 2 u−w? Compute the second component only: 2(−2) minus the second component of w. Apply the scalar before subtracting the vector. Substitute the given numbers: 2 × (-2)-5 = -9. Therefore the requested value is -9.
Contrast a plausible mistake
For that example, the tempting answer -7 follows this error: Forget to multiply the second component of u by 2. The requested quantity is -9. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
A=[[3,2],[−3,5]] and B=[[1,4],[5,−2]]. What is entry (2,2) of 3 A−B? Entry (2,2) is in the second row and second column. Scale that entry of A by 3, then subtract the corresponding negative entry of B. Substitute the given numbers: 3 × 5-(-2) = 17. Therefore the requested value is 17.
Transfer before checking
Vectors u=(4,−2) and w=(3,6) are given. What is the second component of 2 u−w? Try without viewing the answer. Check: -10. Compute the second component only: 2(−2) minus the second component of w. Apply the scalar before subtracting the vector. Substitute the given numbers: 2 × (-2)-6 = -10.
Scope and conditions
Vector components can be negative although magnitude cannot. A determinant can be negative or zero. Do not replace these different quantities by a row-column product. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
Vectors u=(3,−2) and w=(3,5) are given. What is the second component of 2 u−w?
- Compute the second component only: 2(−2) minus the second component of w.
- Apply the scalar before subtracting the vector.
- Substitute the given numbers: 2 × (-2)-5 = -9.
- Therefore the requested value is -9.
Compare the other answer choices
Report the first component instead of the requested second component.
Forget to multiply the second component of u by 2.
Multiply the subtracted component of w by 2 as well.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content