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Scalars and vectors: a geometric approach

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Scalars and vectors: a geometric approach

Vectors have proved to be an effective tool in solving some of the otherwise arduous problems in physics. In this article we learn a way to visualize them. And

As we explained in our previous article, vectors are quantities that not only have a magnitude but also have a specific direction associated with them. This interpretation of vectors becomes much more clear and obvious if we use arrows to represent them. An arrow, by its very definition, has a specific length, and always seems to be pointing in a particular direction. Hence a quantity that requires the specification of both its magnitude and its direction (a vector), can easily be represented with arrows, with the length of the arrow being proportional to the vector’s magnitude and the direction of the arrowhead being the direction of the vector. The arrow-representation of vectors makes it much more easier to visualize them and also enables us to see the various operations performed on them, with ease.

Vector arrow

Representation of a vector by an arrow

Vector operations

Scalar multiplication

A vector can be multiplied with a “scalar” to obtain a new vector. The nature of this new vector depends on the magnitude and sign of the multiplied scalar. If the scalar were greater than 1 (and hence positive), the new resultant vector will be longer than the original vector but will point in the same direction (as the first vector). If the scalar were less than -1 (and hence negative), the new vector will again be correspondingly longer, but will point in a direction exactly opposite to the original. And, if the magnitude of the scalar lies somewhere between -1 and 1, the new vector will shrink in length correspondingly, with its direction (as before) depending on the sign of the scalar.

The long and the short of it is – multiply a vector by a scalar, and you get a new vector whose magnitude is simply the product of the magnitude of the original vector and that of the multiplied scalar (without the sign), and the direction of the new vector having flipped by 180 degrees only if the scalar multiplied is negative.

Scalar multiplication

Observe the effect, scalar multiplication has on a given vector A

Addition of vectors

Algebraically, adding 2 vectors means – adding their corresponding components to get the resultant vector. But, geometrically, there’s a beautiful way to add vectors described by the parallelogram law of addition. To understand the law, consider 2 arbitrary vectors A and B, pointing in different directions as shown below.

 

Vectors

The two arbitrary vectors A and B

To add the vectors A and B, we simply have to place them in a manner so that their tails touch. Now, considering A and B to be 2 sides of a parallelogram, we complete the parallelogram by drawing its other 2 sides, as you can see below.

 

Parallelogram law of vector addition

Completing the parallelogram

Now draw a vector C along a diagonal of the parallelogram, form the point O to the opposite vertex P. This vector C is the sum of the given vectors A and B. To verify this result, simply note down the components of the resultant by introducing a coordinate system, and compare them with the sum of the corresponding coordinates of the vectors A and B, in the same coordinate system.

 

Parallelogram law of vector addition

Finding the resultant using the parallelogram law

An equivalent way to add vector is to use the triangle law of addition. According to this law, in order to obtain the sum of two vectors, you must place them so that the tail of one of them touches the head of the other, and then completing a triangle by drawing a vector from the tail of the first vector to the head of the second, to obtain the resultant. The illustration below will make the process clearer to you.

 

Triangle law of vector addition

Adding vectors with the triangle law

Realize however that the two laws – the parallelogram and the triangle law of addition – are equivalent. What was the diagonal of the parallelogram in the first law, is the hypotenuse of the triangle in the second law.

Extending the vector addition to more than 2 vectors is straightforward, and involves the application of the vector addition laws required number of times until the resultant is obtained.

Vector addition

Considering 3 vectors A, B and C, we first find the resultant D of A and B, then add D to the third vector C, to obtain the final resultant E which is the sum of A, B and C.

Subtracting vectors

Once you’ve learnt scalar multiplication and vector addition; vector subtraction is simply the simultaneous application of the 2 operations. If two vectors, say A and B, are to be subtracted, the resultant will be a new vector, – B. However, instead of defining a new law for vector subtraction, we can view the subtraction as a 2-step process – firstly, multiply the vector B by the scalar -1, which, as you’d know by now, will result in a vector of the same length as B but pointing in the exact opposite direction (written a –B); and then add –B to the vector A using the triangle law of vector addition; yielding+ (-B) = – B, the required difference of A and B.

 

Vector inversion

Inverting the vector B to perform vector subtraction of vectors A and B

Vector subtraction

Adding vectors A and -B, an operation equivalent to subtracting A and B

In this article, we have learnt the geometric representation of vectors and hence have realized a method to visualize the various operations performed on them.

Cover image: Steve Snodgrass

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Roshan Sawhil is a Physics postgraduate who rejoices both doing and explaining Physics. He also finds doing Philosophy as a leisure activity quite interesting. You can find and connect with him on Facebook and Twitter.

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  1. Pingback: Operations on scalars and vectors, part 1Physics Capsule

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