Solving Vector Problems

A final word on notation; in type, vectors are indicated by bold type.In handwriting, it is a convention to underline vectors and leave scalars (such as the constants $k$ and $\lambda$ above without underlining.Remember, $\mathbf.\mathbf=|a|^2$, and if two vectors are perpendicular, their scalar product is

Vector equation of a line Some students are intimidated by the vector equation of a line when they first meet it.(moderate) Two displacements with magnitudes of 10 m and 12 m can be combined to form resultant vectors with many different magnitudes.Which of the following magnitudes can result from these two displacments? For the possible resultants, what angle exists between the original displacements? (moderate) A bicycle tire (Radius = R = 0.4 m) rolls along the ground (with no slipping) through three-quarters of a revolution.Consider the point on the tire that was originally touching the ground.How far has it displaced from its starting position? (moderate) A student carries a lump of clay from the first floor (ground level) door of a skyscraper (on Grant Street) to the elevator, 24 m away. Finally, she exits the elevator and carries the clay 12 m back toward Grant Street.Many students are often reluctant to tackle questions using vectors.I think this is partly because often vectors is not taught until quite a way through a school maths course, so they are unfamiliar.This short article aims to highlight some of the powerful techniques that can be used to solve problems involving vectors, and to encourage you to have a go at such problems to become more familiar with vector properties and applications. When we first meet them, it's often in the context of transformations - a translation can be expressed as a vector telling us how far something is translated to the right (or left) and up (or down).Confusion can strike when we come across vectors being used to indicate absolute position relative to an origin as well as showing a direction.For me, diagrams make it much easier to make sense of what is going on - I can represent a position vector as a point on the diagram with a line segment coming from the origin.Direction vectors just become line segments joined onto other vectors, with a helpful arrow to remind me that $\mathbf$ and $\mathbf$ are in opposite directions!

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Vector equation of a line Some students are intimidated by the vector equation of a line when they first meet it.

(moderate) Two displacements with magnitudes of 10 m and 12 m can be combined to form resultant vectors with many different magnitudes.

Which of the following magnitudes can result from these two displacments? For the possible resultants, what angle exists between the original displacements? (moderate) A bicycle tire (Radius = R = 0.4 m) rolls along the ground (with no slipping) through three-quarters of a revolution.

Consider the point on the tire that was originally touching the ground.

How far has it displaced from its starting position? (moderate) A student carries a lump of clay from the first floor (ground level) door of a skyscraper (on Grant Street) to the elevator, 24 m away. Finally, she exits the elevator and carries the clay 12 m back toward Grant Street.

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Vector equation of a line Some students are intimidated by the vector equation of a line when they first meet it.(moderate) Two displacements with magnitudes of 10 m and 12 m can be combined to form resultant vectors with many different magnitudes.Which of the following magnitudes can result from these two displacments? For the possible resultants, what angle exists between the original displacements? (moderate) A bicycle tire (Radius = R = 0.4 m) rolls along the ground (with no slipping) through three-quarters of a revolution.Consider the point on the tire that was originally touching the ground.How far has it displaced from its starting position? (moderate) A student carries a lump of clay from the first floor (ground level) door of a skyscraper (on Grant Street) to the elevator, 24 m away. Finally, she exits the elevator and carries the clay 12 m back toward Grant Street.Many students are often reluctant to tackle questions using vectors.I think this is partly because often vectors is not taught until quite a way through a school maths course, so they are unfamiliar.This short article aims to highlight some of the powerful techniques that can be used to solve problems involving vectors, and to encourage you to have a go at such problems to become more familiar with vector properties and applications. When we first meet them, it's often in the context of transformations - a translation can be expressed as a vector telling us how far something is translated to the right (or left) and up (or down).Confusion can strike when we come across vectors being used to indicate absolute position relative to an origin as well as showing a direction.For me, diagrams make it much easier to make sense of what is going on - I can represent a position vector as a point on the diagram with a line segment coming from the origin.Direction vectors just become line segments joined onto other vectors, with a helpful arrow to remind me that $\mathbf$ and $\mathbf$ are in opposite directions!

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