Which of the following pairs of vectors are orthogonal?
(i) v = 18i − 3j and w = −i − 6j
(ii) v = 15i − 2j and w = −4i + 30j
(iii) v = 3i − j and w = i + 3j
A. (i) and (ii)
B. (i) and (iii)
C. (ii) only
D. (i) only
17.
Calculate the dot product of the three pairs of vectors. If the dot product is zero, the vectors are orthogonal.
For example,
(i)<18,-3>.<-1,-6>=-18+18=0
So pair (i) is orthogonal.
Post your answer for checking if you wish.
v•w = 0 if v┴w
18i−3j • −i−6j = 0
15i−2j • −4i+30j = -120
3i−j • i+3j = 0
Can you take it from here?
To determine if two vectors are orthogonal, we need to check if their dot product is zero.
The dot product of two vectors v = (v1, v2) and w = (w1, w2) is given by the formula:
v · w = (v1 * w1) + (v2 * w2)
Let's calculate the dot products for the given pairs of vectors:
(i) v = 18i - 3j and w = -i - 6j:
v · w = (18 * -1) + (-3 * -6) = -18 + 18 = 0
(ii) v = 15i - 2j and w = -4i + 30j:
v · w = (15 * -4) + (-2 * 30) = -60 - 60 = -120
(iii) v = 3i - j and w = i + 3j:
v · w = (3 * 1) + (-1 * 3) = 3 - 3 = 0
From the calculations, we can see that the dot product is zero for pairs (i) and (iii).
Therefore, the correct answer is B. (i) and (iii)
To determine whether two vectors are orthogonal, we need to calculate their dot product.
The dot product of two vectors v = (v1, v2) and w = (w1, w2) is given by the formula:
v · w = v1 * w1 + v2 * w2
If the dot product is equal to zero, then the vectors are orthogonal.
Now let's calculate the dot product for each pair of vectors:
(i) v = 18i - 3j and w = -i - 6j:
v · w = (18 * -1) + (-3 * -6) = -18 + 18 = 0
Therefore, v and w are orthogonal.
(ii) v = 15i - 2j and w = -4i + 30j:
v · w = (15 * -4) + (-2 * 30) = -60 - 60 = -120
Therefore, v and w are not orthogonal.
(iii) v = 3i - j and w = i + 3j:
v · w = (3 * 1) + (-1 * 3) = 3 - 3 = 0
Therefore, v and w are orthogonal.
So, the pair of vectors that are orthogonal are (i) and (iii).
The answer is B. (i) and (iii).