For what value or values) of q is the vector A= hat iq +3 hat j + hat k perpendicular to the vector
B= hat iq -q hat j +2 hat k
To find the value(s) of q for which vector A is perpendicular to vector B, we need their dot product to be zero.
The dot product of two vectors A = Ai + Aj + Ak and B = Bi + Bj + Bk is given by:
A • B = (Ai * Bi) + (Aj * Bj) + (Ak * Bk)
In this case:
A = iq + 3j + k,
B = iq - qj + 2k.
Taking the dot product:
A • B = (iq * iq) + (3j * -qj) + (k * 2k)
= q^2 + (-3q^2) + 2
= -2q^2 + 2
To find the value(s) of q for which A • B = 0:
-2q^2 + 2 = 0.
Solving this equation:
-2q^2 = -2
q^2 = 1
q = ±1.
Therefore, the value(s) of q for which vector A is perpendicular to vector B is q = ±1.
To find the value(s) of q for which vector A is perpendicular to vector B, we need to ensure that their dot product is zero.
The dot product of two vectors A and B is given by:
A · B = (Ai * Bi) + (Aj * Bj) + (Ak * Bk)
For our case, we have vector A = qi + 3j + k and vector B = qi - qj + 2k.
Let's calculate their dot product:
A · B = (qi * qi) + (3 * -q) + (1 * 2)
Simplifying this equation:
A · B = q^2 - 3q + 2
To ensure that A is perpendicular to B, their dot product must be zero:
0 = q^2 - 3q + 2
Now, let's solve this quadratic equation to find the values of q:
q^2 - 3q + 2 = 0
Using factoring or the quadratic formula, we can solve for q:
(q - 2)(q - 1) = 0
This equation has two solutions:
q - 2 = 0
q = 2
q - 1 = 0
q = 1
So, the values of q for which vector A is perpendicular to vector B are q = 2 and q = 1.
To determine the value(s) of q for which vector A is perpendicular to vector B, we need to calculate their dot product. If the dot product of two vectors is zero, it means they are perpendicular.
The dot product of two vectors A and B is given by the formula:
A · B = AiBi + AjBj + AkBk
In this case, let's calculate the dot product of A and B:
A · B = (i)(i)q + (3)(-q) + (1)(2)
= q - 3q + 2
= -2q + 2
To make A perpendicular to B, their dot product must be zero. So we set -2q + 2 equal to zero and solve for q:
-2q + 2 = 0
-2q = -2
q = -2 / -2
q = 1
Therefore, when q = 1, the vectors A and B are perpendicular to each other.