Scalar Product of Two Vectors
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Questions and Answers

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

  • 30
  • 50
  • 45 (correct)
  • 40

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

  • θ (correct)
  • 180 degrees
  • 0 degrees
  • 90 degrees

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

  • 10
  • 14
  • 16
  • 12 (correct)

    • What is the characteristic of a scalar quantity?

    <p>It has magnitude but no direction</p> Signup and view all the answers

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

    <p>20</p> Signup and view all the answers

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

    <p>0</p> Signup and view all the answers

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

    <p>14</p> Signup and view all the answers

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

    <p>a.b = a1 b1 + a2 b2 + a3 b3</p> Signup and view all the answers

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

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

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

    <p>m = (y2 - y1) / (x2 - x1)</p> Signup and view all the answers

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

    <p>y - y1 = (y2 - y1) / (x2 - x1)</p> Signup and view all the answers

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

    <p>y = 5/3x - 38</p> Signup and view all the answers

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

    <p>-38</p> Signup and view all the answers

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

    <p>y = mx + c</p> Signup and view all the answers

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

    <p>m = undefined</p> Signup and view all the answers

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

    <p>y = 5/3x - 38</p> Signup and view all the answers

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

    <p>0</p> Signup and view all the answers

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

    <p>The smaller angle having as sides the half-lines starting from the intersection point of the lines</p> Signup and view all the answers

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

    <p>The gradient of a line is equal to the tangent of the angle it makes with the x-axis</p> Signup and view all the answers

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

    <p>tan β = (m2 - m1) / (1 + m1 m2)</p> Signup and view all the answers

    What is the condition for two lines to be parallel?

    <p>The gradients of the two lines are equal</p> Signup and view all the answers

    What is the condition for two lines to be perpendicular?

    <p>1 + m2 m1 = 0</p> Signup and view all the answers

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

    <p>It has no finite value</p> Signup and view all the answers

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

    <p>0</p> Signup and view all the answers

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

    <p>A circle</p> Signup and view all the answers

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

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

    What is the objective of studying coordinate geometry of circles?

    <p>To find the equation of a circle</p> Signup and view all the answers

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

    <p>Blitzer</p> Signup and view all the answers

    What is the concept of touching circles in coordinate geometry?

    <p>When two circles are tangent to each other</p> Signup and view all the answers

    What is the parametric equation of a circle?

    <p>x = rcos(θ), y = rsin(θ)</p> Signup and view all the answers

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

    <p>y = (y2 - y1) / (x2 - x1) (x - x1) + y1</p> Signup and view all the answers

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

    <p>Finding the angle between two lines</p> Signup and view all the answers

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

    <p>The number of radius measures</p> Signup and view all the answers

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

    <p>a is the horizontal radius and b is the vertical radius</p> Signup and view all the answers

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

    <p>x^2/a^2 + y^2/b^2 = 1</p> Signup and view all the answers

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

    <p>To translate the ellipse to the origin</p> Signup and view all the answers

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

    <p>2a</p> Signup and view all the answers

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

    <p>a &gt; b &gt; 0</p> Signup and view all the answers

    What are the coordinates of the foci of an ellipse?

    <p>(±c, 0)</p> Signup and view all the answers

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

    <p>(x-a)^2/a^2 + (y-b)^2/b^2 = 1</p> Signup and view all the answers

    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.

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