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.
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:
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