let a = ci + j and b= 4i + 3j. Find c so that and b are orthogonal
what is the value of c?
To find the value of c such that a and b are orthogonal, we need to calculate the dot product between a and b and set it equal to zero. The dot product of two vectors is given by the sum of the products of their corresponding components.
Given a = ci + j and b = 4i + 3j, we can calculate the dot product as follows:
a · b = (ci + j) · (4i + 3j)
= c(4i) + c(3j) + 1(i) + 1(3j)
= 4ci + 3cj + i + 3j
For a and b to be orthogonal, their dot product must be zero:
4ci + 3cj + i + 3j = 0
Now, let's separate the terms with i and j:
(4c + 1)i + (3c + 3)j = 0
For this equation to hold true, the coefficients of i and j must both be zero:
4c + 1 = 0 => 4c = -1 => c = -1/4
3c + 3 = 0 => 3c = -3 => c = -1
Hence, c = -1/4 is the value that makes a and b orthogonal.
Two vectors are orthogonal if their dot product is zero.
The dot product of two vectors a and b is given by the formula: a · b = (a1 * b1) + (a2 * b2).
Given vectors a = ci + j and b = 4i + 3j, we can find their dot product:
a · b = (c * 4) + (1 * 3)
= 4c + 3
For a and b to be orthogonal, their dot product should be zero:
4c + 3 = 0
Now we solve this equation for c:
4c = -3
c = -3/4
Therefore, the value of c that makes a and b orthogonal is c = -3/4.
orthogonal means they are perpendicular, that is, their dot product is zero
(c,1) dot (4,3) = 0
4c + 3 = 0
c = -3/4