How to Add and Subtract Vectors

To add and subtract vectors that are quantities having magnitude and direction, there are special procedures. You can use the graphical method or the vector components to add and subtract vectors. This article explains both methods. First, let us see how to add vectors graphically.

How to Add Vectors Graphically

Suppose we need to add the two vectors,  \vec{a} and \vec{b}

Vector Addition

Graphically, we place the “tail” of one of the vectors on the “nose” of the other vector. The resultant vector, i.e. the vector  \vec{a}+\vec{b}, is the vector that goes from the tail of the first vector to the nose of the last vector. Note that we can repeat this procedure to add any number of vectors. In practice, to do this, one may need to make a scale diagram of the vectors on a paper.

Adding vectors graphically

Vector addition is commutative, i. e. \vec{a}+\vec{b}=\vec{b}+\vec{a}. Graphically, we can show this by starting with \vec{b} and then adding \vec{a} to it. Since the resultant vector in both cases have the same size and direction, the resultant vectors are equal. This shows that the order in which vectors are added does not matter.

Vector addition is commutative

Also, vector addition is associative, i.e. \vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}. This means that the order in which vectors are grouped during addition does not matter.

How to Add Vectors Using Components

Suppose the vector \vec{a}=(a_x, a_y, a_z) and the vector \vec{b}=(b_x, b_y, b_z). Then, the vector \vec{a}+\vec{b}=(a_x+b_x, a_y+b_y, a_z+b_z).


Two forces \vec{F_1} and \vec{F_2} act on a body. The forces are given by the vectors \vec{F_1} = (2, 8, -5) \textup{N} and \vec{F_2} = (-1, 3, 3) \textup{N}. Find the resultant force F_R.

F_R = (2+(-1), 8+3, (-5)+3) = (1, 11, -2) \textup{N}.

Similarly, subtracting vectors involves adding the negative of a vector.

Negative of a Vector

For a given vector \vec{p}, the negative vector \vec{-p} has the same magnitude as \vec{p} but points in the opposite direction.

The negative of a vector has the same magnitude, but opposite direction

How to Subtract Vectors Graphically

Subtracting a vector is equivalent to adding the negative vector. i.e. \vec{a}-\vec{b} = \vec{a}+(-\vec{b}). Graphically, we can show this as follows:

We need to subtract \vec{b} from \vec{a}.

Vector Addition

We identify the negative vector of \vec{b}. Then, \vec{a}-\vec{b} = \vec{a}+(-\vec{b}):

How to Add and Subtract Vectors

How to Subtract Vectors Using Components

Suppose the vector \vec{a}=(a_x, a_y, a_z) and the vector \vec{b}=(b_x, b_y, b_z). Then, the vector \vec{a}-\vec{b}=(a_x-b_x, a_y-b_y, a_z-b_z).

For example, the displacement of a particle can be found by subtracting two position vectors. For worked example, see the article on displacement.

How to Multiply Vectors

About the Author: Nipun

Related pages

bullying vs teasingmetallic bondcytosol cytoplasmmandeleev periodic tablesensory imagery examplesdifference between risk threat and vulnerabilitycharacteristics of asexual reproductiondifference between food poisoning and the fluidentify adjectiveintrons and exons definitionexamples for homographsdifference between mitochondria and mitochondriondifference between internal and external fertilisationhow does an electromagnet differ from a permanent magnetinvoke and evokecompare static and current electricitydefine spectrophotometrysign and symptoms of hypoglycemia and hyperglycemiadefine cations and anionsconvex and concave lensedefine pathos in literatureenjambment poetryprozac chemical formulacovalent ionic and metallic bondingnz flag vs australian flagdifference between permeable and impermeablewhat is the difference between prince and dukenames of essential and nonessential amino acidswhatisadverbsherbert and sorbetdifference between cold sores and fever blisterssuperego definewhat is the meaning of the word bewilderedmeaning of patronising someonedifference between vernier caliper and micrometer screw gaugecell protoplasm is made up mostly ofprojected balance sheet definitionvolatile substances examplesdifference between hispanic and spanishdefinition of anode and cathodeis baking soda the same as bicarbonate sodadifference between a ged and high school diplomadifference between a cyclone and a tornadowhat is the difference between monocots and dicotslevis black label 514what is internal rhymecomparison of technical writing and literary writinghepatitis jaundicedifference in horsepower and torquedefinition heterotrophsalaskan malamute husky differencedioecy definitionatp chemical formuladifferentiate between reflection and refractiondifference between phonology and phoneticsmeaning of anectodedifferences between prokaryote and eukaryoteprologue literary definitionfunctions of mitochondria and chloroplastsxray waveunipolar vs bipolar depressionbrass bronze copper differencedifference between curriculum and syllabusshinkansen maglev trainwhat are the different kinds of adjectivedefinition of temporary magnetsuperego psychology definitiondefinition of rhyme and examplestest for so4myth vs legendexamples of conceitsdifference adverb and adjectivewhat is simile metaphorwhat's the definition of cytoplasmsuccessor and predecessor definition