Let be a line given by A + tB and let P be a point on different from A. For point Q not on , show that the distance d of Q form the line is given by:
To find the distance between point Q and the line , we can use the formula for the distance between a point and a line in three-dimensional space.
The formula is:
d = |(QP) x B| / |B|
Where QP is the vector from point Q to any point on the line (such as A), x denotes the cross product, and | | denotes the magnitude of a vector.
Let's calculate the distance using this formula.
Step 1: Find vector QP
We can calculate vector QP by subtracting the coordinates of point Q from the coordinates of point A:
QP = A - Q
Step 2: Calculate the cross product (QP) x B
To calculate the cross product, we can write the vectors QP and B as column vectors and take their determinant:
(QP) x B = | i j k |
| QPx QPy QPz|
| Bx By Bz |
The components of vector (QP) x B will be the determinants of the 2x2 submatrices formed by deleting the row and column of each component. We'll get:
(QP) x B = (QPy * Bz - QPz * By) i + (QPz * Bx - QPx * Bz) j + (QPx * By - QPy * Bx) k
Step 3: Calculate the magnitude of vector (QP) x B
To calculate the magnitude of the cross product vector, we can use the formula:
|(QP) x B| = sqrt((QPy * Bz - QPz * By)^2 + (QPz * Bx - QPx * Bz)^2 + (QPx * By - QPy * Bx)^2)
Step 4: Calculate the magnitude of vector B
The magnitude of vector B is given by:
|B| = sqrt(Bx^2 + By^2 + Bz^2)
Step 5: Calculate the distance d
Using the values calculated in steps 3 and 4, we can now calculate the distance:
d = |(QP) x B| / |B|
And that's the formula for the distance between a point and a line in three-dimensional space.
To find the distance d from a point Q to a line , we can use the following formula:
d = ||(A - Q) - ((A - Q) · B) * B|| / ||B||
Here, "|| ||" denotes the magnitude (or length) of a vector, "·" denotes the dot product of two vectors, A and B are position vectors that define the line , and Q is the given point.
Now, let's work step-by-step to demonstrate how the formula is derived and compute the distance d.
Step 1: Find a vector perpendicular to the line .
We know that any vector perpendicular to the line can be found by taking the cross product of the direction vector B and any other non-zero vector that is not parallel to B. For simplicity, let's choose a random vector C that is not parallel to B.
Step 2: Define the vectors from point A, point Q, and point P to calculate the distance.
Let R = (A - Q) be the vector from point Q to point A.
Let S = (A - Q) · B be the scalar projection of R onto B.
Let T = S * B be the vector projection of R onto B.
Step 3: Express the vector T using vector C.
By definition, T is parallel to B. Therefore, T can be written as T = k * C, where k is a constant.
Step 4: Find the value of the constant k.
Since T is parallel to B, we can take the dot product of T and B to get:
T · B = (k * C) · B = k * (C · B)
Step 5: Combine the equations to solve for k.
We have two expressions for T · B:
T · B = (A - Q) · B
T · B = k * (C · B)
Setting the two expressions equal to each other, we get:
(A - Q) · B = k * (C · B)
Now solve for k:
k = [(A - Q) · B] / (C · B)
Step 6: Calculate the vector T using k.
Substituting the value of k back into T = k * C, we obtain:
T = [(A - Q) · B / (C · B)] * C
Step 7: Calculate the distance using the formula.
Finally, we can calculate the distance d from the point Q to the line :
d = ||(A - Q) - T||
= ||(A - Q) - [(A - Q) · B / (C · B)] * C||
= ||(A - Q) - [(A - Q) · B] / (C · B) * C||
And that is the formula for the distance d from the point Q to the line .
Note: Remember to choose the vector C that is not parallel to B in step 1.