1. Consider the two vectors P and Q. The definition of P is P = i × (i × j). The
definition of Q is Q = (i × i) × j .
a. The magnitude of P is equal to the magnitude of Q.
b. The magnitude of P is greater than the magnitude of Q.
c. The magnitude of P is less than the magnitude of Q.
Is the answer A
P
(i x j) = k
i x k = -j
Q
i x i = 0
0 x j = 0
so|P| > |Q|
To determine whether the answer is A, B, or C, we need to find the magnitudes of vectors P and Q and compare them.
Let's start by finding the magnitude of vector P. The definition of P is P = i × (i × j).
To find the magnitude of P, we can use the properties of vector cross product:
P = i × (i × j)
P = i × (j × i)
(Note: the order of the vectors in the cross product matters)
Now, we can use the rule for calculating the cross product:
P = (i × j) × i
P = k × i
(Note: the cross product of i and j is k)
Since the cross product of k and i is not zero, we know that the magnitude of vector P is not zero.
Now, let's find the magnitude of vector Q. The definition of Q is Q = (i × i) × j.
To find the magnitude of Q, we can again use the properties of vector cross product:
Q = (i × i) × j
(Note: the order of the vectors in the cross product matters)
Now, we can use the rule for calculating the cross product:
Q = 0 × j
(Note: the cross product of i and itself is zero)
Since the cross product of 0 and j is zero, we know that the magnitude of vector Q is zero.
Comparing the magnitudes of P and Q, we can conclude that the magnitude of P is greater than the magnitude of Q. Therefore, the answer is B: "The magnitude of P is greater than the magnitude of Q."