## Podcast Beta

## Questions and Answers

What is the scalar product of two vectors a and b when a = 3i + 4j + 5k and b = 2i + j + 7k?

What is the angle between two vectors a and b if their scalar product is equal to ab?

If a = 2i + 2j - k and b = 3i - 6j + 2k, what is the scalar product of a and b?

What is the characteristic of a scalar quantity?

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If a = 5i + 4j + 2k, b = 4i - 5j + 3k, and c = 2i - j - 2k, what is the value of a.b?

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What is the dot product of two vectors at right angles to each other?

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If a = 2i + 4j - 3k and b = i + 3j + 2k, what is the scalar product of a and b?

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What is the formula for the scalar product of two vectors a and b in terms of unit vectors?

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What is the equation of a straight line passing through a point p (x1, y1) with a gradient m?

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What is the formula to find the gradient of a line passing through two points (x1, y1) and (x2, y2)?

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What is the equation of a straight line passing through two points (x1, y1) and (x2, y2)?

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If a line has a gradient of 5/3 and passes through the point (1,3), what is its equation?

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What is the intercept of the line y = 5/3x - 38 on the y-axis?

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What is the form of the equation of a straight line?

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What is the gradient of a line passing through two points (x1, y1) and (x2, y2) if the points are equal?

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What is the equation of a straight line passing through the point (1,4) with a gradient of 5/3?

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What is the angle between two lines in a plane if they are parallel?

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What is the smaller angle between two lines in a plane, if the lines are not parallel?

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What is the relationship between the gradient of a line and the angle it makes with the x-axis?

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What is the formula for the tangent of the acute angle between two lines?

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What is the condition for two lines to be parallel?

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What is the condition for two lines to be perpendicular?

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What is the value of tan β when the lines are perpendicular?

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What is the value of tan β when the lines are parallel?

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What is the locus of all points equidistant from a central point?

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What is the equation of a circle with center at the origin and radius r?

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What is the objective of studying coordinate geometry of circles?

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What is the name of the author who wrote the book 'Algebra and Trigonometry Custom'?

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What is the concept of touching circles in coordinate geometry?

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What is the parametric equation of a circle?

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What is the equation of a line that passes through two points?

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What is the concept of finding the acute angle between two lines?

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What is the primary difference between a circle and an ellipse?

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What is the convention for labeling the radius measures in an ellipse?

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What is the equation of an ellipse centered at the origin?

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What is the purpose of subtracting offsets from the x and y terms in the equation of an ellipse?

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What is the length of the major axis of an ellipse?

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What is the condition for the equation of an ellipse to be true?

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What are the coordinates of the foci of an ellipse?

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What is the equation of an ellipse that is not centered at the origin?

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## Study Notes

### Scalar Product of Two Vectors

- The scalar product (dot product) of two vectors
`a`

and`b`

is denoted by`a.b`

and is equal to`ab cos θ`

, where`θ`

is the angle between`a`

and`b`

. - If
`a = a1 i + a2 j + a3 k`

and`b = b1 i + b2 j + b3 k`

, then`a.b = a1 b1 + a2 b2 + a3 b3`

. - The dot product of two vectors at right angles to each other is zero.

### Exercises

- Find the scalar product of two vectors
`a`

and`b`

given their components. - Determine the value of
`a.b`

,`a.c`

, and`b.c`

given three vectors`a`

,`b`

, and`c`

.

### Vector Product of Two Vectors

- The vector product is a method for constructing a vector perpendicular to a plane if you have two vectors in the plane.
- The vector product has numerous applications in physics and astronomy.

### LOCI (Lines and Coordinate Geometry)

- The equation of a straight line passing through a point
`(x1, y1)`

with a gradient`m`

is`y - y1 = m(x - x1)`

. - The equation of a straight line passing through two points
`(x1, y1)`

and`(x2, y2)`

is`y - y1 = (y2 - y1)/(x2 - x1)(x - x1)`

. - The angle between two lines is defined as 0 if the lines are parallel, or the smaller angle between the two lines if they are not parallel.

### Angle Between Two Lines

- The acute angle between two lines with gradients
`m1`

and`m2`

is given by`tan β = (m2 - m1)/(1 + m1 m2)`

, where`β`

is the acute angle between the lines.

### Parallel Lines

- If two lines are parallel, then their gradients are equal, i.e.,
`m1 = m2`

.

### Perpendicular Lines

- If two lines are perpendicular, then the product of their gradients is -1, i.e.,
`m1 m2 = -1`

.

### Exercises

- Find the equation of a line passing through two points.
- Find the equation of a line parallel to a given line and passing through a point.
- Find the acute angle between two lines.

### Coordinate Geometry (Circle)

- A circle is the locus of all points equidistant from a central point.
- The equation of a circle with center at the origin and radius
`r`

is`x^2 + y^2 = r^2`

.

### Equation of a Circle

- The equation of a circle with center at
`(h, k)`

and radius`r`

is`(x - h)^2 + (y - k)^2 = r^2`

.

### Ellipse and Hyperbola

- The equation of an ellipse centered at the origin is
`x^2/a^2 + y^2/b^2 = 1`

, where`a`

is the radius along the x-axis and`b`

is the radius along the y-axis. - The equation of an ellipse not centered at the origin is
`(x - h)^2/a^2 + (y - k)^2/b^2 = 1`

, where`(h, k)`

is the center of the ellipse. - The length of the major axis of an ellipse is
`2a`

and the length of the minor axis is`2b`

.

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## Description

Learn about the concept of scalar product of two vectors and its expression in terms of unit vectors. Practice calculating scalar product with examples.